Matrix Operations | Matrices | Linear Algebra

1. Matrix Addition and Scalar Multiplication

Before studying matrix multiplication, we define the simpler operations that make the space of matrices into a vector space.

Definition 4.3: Matrix Addition

Let $A = (a_{ij})$ and $B = (b_{ij})$ be $m \times n$ matrices over $F$ . Their sum is:

A + B = (a_{ij} + b_{ij})_{m \times n}

Addition is performed entry-by-entry. Matrices must have the same dimensions to be added.

Definition 4.4: Scalar Multiplication

For $\lambda \in F$ and $A = (a_{ij})$ :

\lambda A = (\lambda a_{ij})_{m \times n}

Each entry is multiplied by the scalar.

Theorem 4.5: Vector Space Structure

The set $F^{m \times n}$ of all $m \times n$ matrices over $F$ forms a vector space with:

Zero element: the zero matrix $O = (0)_{m \times n}$
Additive inverse: $-A = (-a_{ij})$
Dimension: $\dim F^{m \times n} = mn$

Remark 4.7: Connection to Linear Maps

These operations correspond to operations on linear maps. If $A$ represents $\sigma$ and $B$ represents $\tau$ , then:

$A + B$ represents $\sigma + \tau$
$\lambda A$ represents $\lambda \sigma$

2. Matrix Multiplication

Matrix multiplication is defined to match composition of linear maps. This seemingly complex definition is exactly what makes $[S \circ T] = [S][T]$ work.

Definition 4.5: Matrix Multiplication

Let $A = (a_{ij})_{p \times m}$ and $B = (b_{ij})_{m \times n}$ . Their product $C = AB$ is a $p \times n$ matrix with:

c_{ij} = \sum_{k=1}^{m} a_{ik} b_{kj} = a_{i1}b_{1j} + a_{i2}b_{2j} + \cdots + a_{im}b_{mj}

The $(i,j)$ entry is the dot product of row $i$ of $A$ with column $j$ of $B$ .

Remark 4.8: Dimension Requirements

For $AB$ to be defined:

Columns of $A$ must equal rows of $B$
If $A$ is $p \times m$ and $B$ is $m \times n$ , then $AB$ is $p \times n$
Memory aid: $(p \times \cancel{m})(\cancel{m} \times n) = p \times n$

Theorem 4.6: Multiplication as Composition

Let $\sigma \in L(V_1, V_2)$ and $\tau \in L(V_2, V_3)$ with matrices $B$ and $A$ respectively. Then:

M(\tau \circ \sigma) = M(\tau) \cdot M(\sigma) = AB

Proof:

Let $B_1, B_2, B_3$ be bases for $V_1, V_2, V_3$ . For basis vector $\varepsilon_j \in B_1$ :

(\tau \circ \sigma)(\varepsilon_j) = \tau(\sigma(\varepsilon_j)) = \tau\left(\sum_{k=1}^m b_{kj} \zeta_k\right) = \sum_{k=1}^m b_{kj} \tau(\zeta_k)

= \sum_{k=1}^m b_{kj} \sum_{i=1}^p a_{ik} \eta_i = \sum_{i=1}^p \left(\sum_{k=1}^m a_{ik} b_{kj}\right) \eta_i

The coefficient of $\eta_i$ is $\sum_k a_{ik} b_{kj} = (AB)_{ij}$ .

∎

3. Properties of Matrix Multiplication

Theorem 4.7: Algebraic Properties

Matrix multiplication satisfies:

Associativity: $(AB)C = A(BC)$
Left distributivity: $A(B + C) = AB + AC$
Right distributivity: $(A + B)C = AC + BC$
Scalar compatibility: $\lambda(AB) = (\lambda A)B = A(\lambda B)$
Identity: $AI = IA = A$

Proof:

Associativity: Follows from associativity of function composition: $(\tau \circ \sigma) \circ \rho = \tau \circ (\sigma \circ \rho)$ .

Distributivity: Follows from $\tau \circ (\sigma_1 + \sigma_2) = \tau \circ \sigma_1 + \tau \circ \sigma_2$ .

Alternatively, verify directly using the summation formula for matrix entries.

∎

Remark 4.9: Non-Commutativity

Matrix multiplication is NOT commutative! In general:

AB \neq BA

Several things can happen:

$AB$ exists but $BA$ doesn't (different dimensions)
Both exist but have different dimensions
Both exist with same dimensions but are unequal

Example 4.6: Non-Commutativity

Let $A = \begin{pmatrix} 1 & 1 & -3 \\ 1 & 0 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 3 \\ 1 & 2 \\ 3 & 1 \end{pmatrix}$ .

AB = \begin{pmatrix} -8 & 2 \\ -3 & 2 \end{pmatrix} \quad (2 \times 2)

BA = \begin{pmatrix} 3 & 0 & -3 \\ 3 & 1 & -5 \\ 4 & 3 & -10 \end{pmatrix} \quad (3 \times 3)

Not only are $AB$ and $BA$ different, they have different dimensions!

4. Matrix Transpose

Definition 4.6: Transpose

The transpose of $A = (a_{ij})_{m \times n}$ is $A^T = (a_{ji})_{n \times m}$ :

(A^T)_{ij} = A_{ji}

Rows become columns and columns become rows.

Theorem 4.8: Transpose Properties

For compatible matrices $A$ , $B$ and scalar $\lambda$ :

$(A^T)^T = A$
$(A + B)^T = A^T + B^T$
$(\lambda A)^T = \lambda A^T$
$(AB)^T = B^T A^T$ (order reverses!)

Proof:

Property 4: We verify that $((AB)^T)_{ij} = (B^T A^T)_{ij}$ :

((AB)^T)_{ij} = (AB)_{ji} = \sum_k A_{jk} B_{ki}

(B^T A^T)_{ij} = \sum_k (B^T)_{ik} (A^T)_{kj} = \sum_k B_{ki} A_{jk}

These are equal since multiplication in $F$ is commutative.

∎

Definition 4.7: Symmetric and Skew-Symmetric Matrices

A square matrix $A$ is:

Symmetric if $A^T = A$ (i.e., $a_{ij} = a_{ji}$ )
Skew-symmetric if $A^T = -A$ (i.e., $a_{ij} = -a_{ji}$ )

Theorem 4.9: Symmetric Decomposition

Any square matrix $A$ can be uniquely written as:

A = S + K

where $S$ is symmetric and $K$ is skew-symmetric:

S = \frac{1}{2}(A + A^T), \quad K = \frac{1}{2}(A - A^T)

5. Special Matrices

Definition 4.8: Identity Matrix

The $n \times n$ identity matrix $I_n$ has 1s on the diagonal and 0s elsewhere:

I_n = \begin{pmatrix} 1 & 0 & \cdots & 0 \\ 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & 1 \end{pmatrix}

Property: $AI_n = I_m A = A$ for any $m \times n$ matrix $A$ .

Definition 4.9: Diagonal Matrix

A diagonal matrix has non-zero entries only on the main diagonal:

D = \text{diag}(d_1, d_2, \ldots, d_n) = \begin{pmatrix} d_1 & 0 & \cdots & 0 \\ 0 & d_2 & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & d_n \end{pmatrix}

Theorem 4.10: Diagonal Matrix Properties

For diagonal matrices $D = \text{diag}(d_1, \ldots, d_n)$ and $E = \text{diag}(e_1, \ldots, e_n)$ :

$DE = ED = \text{diag}(d_1 e_1, \ldots, d_n e_n)$
$D^k = \text{diag}(d_1^k, \ldots, d_n^k)$
Diagonal matrices commute with each other

Definition 4.10: Scalar Matrix

A scalar matrix is $\lambda I = \text{diag}(\lambda, \lambda, \ldots, \lambda)$ .

Key property: Scalar matrices commute with ALL matrices of the same size.

Definition 4.11: Triangular Matrices

A matrix is:

Upper triangular if $a_{ij} = 0$ for $i > j$
Lower triangular if $a_{ij} = 0$ for $i < j$

Products of upper (lower) triangular matrices are upper (lower) triangular.

6. Worked Examples

Example 1: Rotation Matrices

Problem: Show that $R_{\theta_1} R_{\theta_2} = R_{\theta_1 + \theta_2}$ .

Solution:

R_{\theta_1} R_{\theta_2} = \begin{pmatrix} \cos\theta_1 & -\sin\theta_1 \\ \sin\theta_1 & \cos\theta_1 \end{pmatrix} \begin{pmatrix} \cos\theta_2 & -\sin\theta_2 \\ \sin\theta_2 & \cos\theta_2 \end{pmatrix}

= \begin{pmatrix} \cos\theta_1\cos\theta_2 - \sin\theta_1\sin\theta_2 & -\cos\theta_1\sin\theta_2 - \sin\theta_1\cos\theta_2 \\ \sin\theta_1\cos\theta_2 + \cos\theta_1\sin\theta_2 & -\sin\theta_1\sin\theta_2 + \cos\theta_1\cos\theta_2 \end{pmatrix}

= \begin{pmatrix} \cos(\theta_1+\theta_2) & -\sin(\theta_1+\theta_2) \\ \sin(\theta_1+\theta_2) & \cos(\theta_1+\theta_2) \end{pmatrix} = R_{\theta_1+\theta_2}

Example 2: Matrix Powers

Problem: Expand $(A + \lambda I)^n$ .

Solution: Since $A$ and $\lambda I$ commute ( $A(\lambda I) = (\lambda I)A = \lambda A$ ), we can use the binomial theorem:

(A + \lambda I)^n = \sum_{k=0}^{n} \binom{n}{k} A^k (\lambda I)^{n-k} = \sum_{k=0}^{n} \binom{n}{k} \lambda^{n-k} A^k

Example 3: Zero Divisors

Problem: Find non-zero matrices $A$ , $B$ with $AB = 0$ .

Solution:

A = \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}

AB = \begin{pmatrix} 1-1 & -1+1 \\ 1-1 & -1+1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

This shows matrices can have zero divisors—unlike real numbers!

Example 4: Column Interpretation

Key insight: Column $k$ of $AB$ equals $A$ times column $k$ of $B$ .

Each column of $AB$ is a linear combination of columns of $A$ :

(AB)_{\cdot k} = b_{1k} A_{\cdot 1} + b_{2k} A_{\cdot 2} + \cdots + b_{mk} A_{\cdot m}

This means the column space of $AB$ is contained in the column space of $A$ .

7. Common Mistakes

Mistake 1: Assuming Commutativity

$AB \neq BA$ in general. Never assume you can swap the order of matrix multiplication. Formulas like $(A + B)^2 = A^2 + 2AB + B^2$ are WRONG for matrices.

Mistake 2: Wrong Dimension Check

For $AB$ , the COLUMNS of $A$ must match ROWS of $B$ . A common error is checking if both matrices are the same size.

Mistake 3: Transpose Order

$(AB)^T = B^T A^T$ , NOT $A^T B^T$ . The order reverses! Same for inverse: $(AB)^{-1} = B^{-1} A^{-1}$ .

Mistake 4: Cancellation

$AB = AC$ does NOT imply $B = C$ . Cancellation fails because matrices can have non-trivial null spaces (zero divisors exist).

Mistake 5: Entry-wise Multiplication

Matrix multiplication is NOT entry-by-entry. The formula involves summing products along rows and columns. Entry-wise product (Hadamard product) is a different operation.

8. Matrix Polynomials

Definition 4.12: Matrix Polynomial

For a polynomial $p(x) = a_m x^m + a_{m-1} x^{m-1} + \cdots + a_1 x + a_0$ and square matrix $A$ :

p(A) = a_m A^m + a_{m-1} A^{m-1} + \cdots + a_1 A + a_0 I

where $A^0 = I$ and $A^k = A \cdot A \cdots A$ ( $k$ times).

Theorem 4.11: Polynomial Commutativity

For any polynomials $f, g$ and matrix $A$ :

f(A) g(A) = g(A) f(A)

Matrix polynomials in the same matrix always commute.

Proof:

Since $A^j A^k = A^{j+k} = A^k A^j$ , powers of $A$ commute. Linearity extends this to all polynomials.

∎

Corollary 4.4: Polynomial Commutativity with Different Matrices

If $AB = BA$ , then $f(A) g(B) = g(B) f(A)$ for any polynomials $f, g$ .

9. Key Takeaways

Row-Column Formula

$(AB)_{ij} = \sum_k A_{ik} B_{kj}$ — row $i$ of $A$ dotted with column $j$ of $B$ .

Composition Connection

$[S \circ T] = [S][T]$ . Matrix multiplication IS composition of linear maps.

Non-Commutativity

$AB \neq BA$ in general. Order matters! Only scalar matrices commute with everything.

Transpose Reversal

$(AB)^T = B^T A^T$ . Transpose and inverse both reverse multiplication order.

10. Quick Reference Summary

Property	Formula
Matrix Product Entry	$(AB)_{ij} = \sum_k A_{ik} B_{kj}$
Associativity	$(AB)C = A(BC)$
Distributivity	$A(B+C) = AB + AC$
Non-Commutativity	$AB \neq BA$ in general
Transpose of Product	$(AB)^T = B^T A^T$
Identity	$AI = IA = A$
Symmetric	$A^T = A$

11. Block Matrices

Definition 4.13: Block Matrix

A block matrix (or partitioned matrix) is a matrix viewed as composed of smaller matrices (blocks):

A = \begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix}

where each $A_{ij}$ is itself a matrix.

Theorem 4.12: Block Multiplication

If $A$ and $B$ are partitioned compatibly, then:

\begin{pmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{pmatrix} \begin{pmatrix} B_{11} & B_{12} \\ B_{21} & B_{22} \end{pmatrix} = \begin{pmatrix} A_{11}B_{11} + A_{12}B_{21} & A_{11}B_{12} + A_{12}B_{22} \\ A_{21}B_{11} + A_{22}B_{21} & A_{21}B_{12} + A_{22}B_{22} \end{pmatrix}

The formula looks like scalar multiplication, but order matters within each term!

Remark 4.10: Compatible Partitioning

For block multiplication to work:

Column partitions of $A$ must match row partitions of $B$
Each block product $A_{ik}B_{kj}$ must be well-defined

Example 4.7: Block Diagonal Matrices

Block diagonal matrices have a particularly simple structure:

\begin{pmatrix} A & 0 \\ 0 & B \end{pmatrix} \begin{pmatrix} C & 0 \\ 0 & D \end{pmatrix} = \begin{pmatrix} AC & 0 \\ 0 & BD \end{pmatrix}

Block diagonal matrices form a subalgebra: sums and products of block diagonal matrices are block diagonal.

12. Row and Column Interpretations

Matrix multiplication can be understood from multiple perspectives, each providing useful insights.

Theorem 4.13: Column View of Matrix Product

If $B = (b_1, b_2, \ldots, b_n)$ where $b_j$ are the columns of $B$ , then:

AB = (Ab_1, Ab_2, \ldots, Ab_n)

Each column of $AB$ is $A$ times the corresponding column of $B$ .

Theorem 4.14: Row View of Matrix Product

If $A = \begin{pmatrix} a_1^T \\ a_2^T \\ \vdots \\ a_p^T \end{pmatrix}$ where $a_i^T$ are the rows of $A$ , then:

AB = \begin{pmatrix} a_1^T B \\ a_2^T B \\ \vdots \\ a_p^T B \end{pmatrix}

Each row of $AB$ is the corresponding row of $A$ times $B$ .

Corollary 4.5: Column Space Inclusion

The column space of $AB$ is contained in the column space of $A$ :

\text{col}(AB) \subseteq \text{col}(A)

Corollary 4.6: Row Space Inclusion

The row space of $AB$ is contained in the row space of $B$ :

\text{row}(AB) \subseteq \text{row}(B)

Theorem 4.15: Outer Product Representation

Matrix multiplication can also be written as a sum of outer products:

AB = \sum_{k=1}^m a_{\cdot k} b_{k \cdot}

where $a_{\cdot k}$ is column $k$ of $A$ and $b_{k \cdot}$ is row $k$ of $B$ .

13. Additional Examples

Example: Nilpotent Matrix

Problem: Show that $N = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}$ is nilpotent.

Solution:

N^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}, \quad N^3 = \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} = O

$N$ is nilpotent of index 3: $N^3 = 0$ but $N^2 \neq 0$ .

Example: Idempotent Matrix

Problem: Show that projection matrices are idempotent ( $P^2 = P$ ).

Solution: For projection onto the line $y = x$ :

P = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix}

P^2 = \begin{pmatrix} 1/4 + 1/4 & 1/4 + 1/4 \\ 1/4 + 1/4 & 1/4 + 1/4 \end{pmatrix} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & 1/2 \end{pmatrix} = P

Projecting twice is the same as projecting once.

Example: Involutory Matrix

Problem: Verify that $A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ is involutory ( $A^2 = I$ ).

Solution:

A^2 = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = I

This is the swap/reflection matrix. Reflecting twice returns to the original.

Example: Matrix Equation

Problem: Solve $A^2 - 3A + 2I = 0$ for information about $A$ .

Solution: Factor: $(A - I)(A - 2I) = 0$ .

This means eigenvalues of $A$ are among $\{1, 2\}$ .

If $A$ is diagonalizable, then $A = PDP^{-1}$ where $D$ has only 1s and 2s on the diagonal.

14. Theoretical Insights

Theorem 4.16: Ring Structure

The set of $n \times n$ matrices $M_n(F)$ forms a ring with:

Additive identity: Zero matrix $O$
Multiplicative identity: Identity matrix $I$
This ring is non-commutative for $n \geq 2$
This ring has zero divisors for $n \geq 2$

Remark 4.11: Differences from Number Rings

Unlike the ring of integers or real numbers:

$AB = 0$ does NOT imply $A = 0$ or $B = 0$
$AB = AC$ does NOT imply $B = C$
$AB = BA$ is NOT generally true
$(A + B)^2 \neq A^2 + 2AB + B^2$ in general

Theorem 4.17: Trace Properties

The trace function $\text{tr}: M_n(F) \to F$ satisfies:

$\text{tr}(A + B) = \text{tr}(A) + \text{tr}(B)$
$\text{tr}(\lambda A) = \lambda \text{tr}(A)$
$\text{tr}(AB) = \text{tr}(BA)$ (trace of product is symmetric!)

Proof:

For the third property:

\text{tr}(AB) = \sum_i (AB)_{ii} = \sum_i \sum_k A_{ik} B_{ki}

\text{tr}(BA) = \sum_i (BA)_{ii} = \sum_i \sum_k B_{ik} A_{ki} = \sum_k \sum_i A_{ki} B_{ik}

Relabeling indices shows these are equal.

∎

15. Challenge Problems

Challenge 1: Trace of Commutator

Prove that $\text{tr}(AB - BA) = 0$ for any $A, B \in M_n(F)$ . Conclude that there are no matrices with $AB - BA = I$ .

Challenge 2: Characterize Commutativity

Prove that if $A$ commutes with every $B \in M_n(F)$ , then $A = \lambda I$ for some $\lambda \in F$ .

Challenge 3: Nilpotent Powers

Let $N$ be nilpotent with $N^k = 0$ . Show that $(I - N)^{-1} = I + N + N^2 + \cdots + N^{k-1}$ .

Challenge 4: Product of Symmetric Matrices

If $A$ and $B$ are symmetric, when is $AB$ symmetric? Prove that $AB$ is symmetric if and only if $AB = BA$ .

16. Connections to Linear Maps

Matrix operations directly correspond to operations on linear maps. This connection is the foundation of computational linear algebra.

Addition: Sum of Maps

If $[\sigma] = A$ and $[\tau] = B$ , then:

[\sigma + \tau] = A + B

Adding matrices corresponds to adding the linear maps they represent.

Multiplication: Composition

If $[\sigma] = B$ and $[\tau] = A$ , then:

[\tau \circ \sigma] = AB

Matrix multiplication IS composition. The non-commutativity reflects that composition of maps is generally non-commutative.

Scalar Multiplication

If $[\sigma] = A$ , then:

[\lambda \sigma] = \lambda A

Scalar multiplication of the matrix corresponds to scalar multiplication of the map.

Invertibility

If $\sigma$ is an isomorphism with $[\sigma] = A$ , then:

[\sigma^{-1}] = A^{-1}

The inverse map corresponds to the inverse matrix.

17. Additional Practice Problems

Problem 1

Compute $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ .

Problem 2

Find matrices $A$ , $B$ such that $AB \neq BA$ but $A + B = B + A$ .

Problem 3

Show that $(A^T)^T = A$ for any matrix $A$ .

Problem 4

If $A^2 = A$ (idempotent), show that $(I - A)^2 = I - A$ .

Problem 5

Prove that if $A$ and $B$ are upper triangular, then $AB$ is upper triangular.

Problem 6

Find the general form of all $2 \times 2$ matrices that commute with $\begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ .

Problem 7

Show that $\text{tr}(A^T A) \geq 0$ for any real matrix $A$ , with equality iff $A = 0$ .

Problem 8

Compute $A^{100}$ where $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ .

18. Study Tips

Check Dimensions First

Before any multiplication, verify dimensions match. $(m \times n)(n \times p) = m \times p$ . The inner dimensions must be equal.

Think Geometrically

Visualize what the matrices do as transformations. Rotation, reflection, scaling—these help build intuition for why certain properties hold.

Use the Column View

Remember: column $j$ of $AB$ is $A$ times column $j$ of $B$ . This simplifies many proofs and computations.

Never Assume Commutativity

Until you prove $AB = BA$ for specific matrices, always preserve order. Most algebraic identities from scalars need modification for matrices.

19. Historical Notes

Cayley and Matrix Multiplication: Arthur Cayley introduced the modern definition of matrix multiplication in 1858. He recognized that it corresponds to composition of linear transformations, which is why the formula is more complex than entry-wise multiplication.

Non-Commutativity: The fact that $AB \neq BA$ was initially surprising to mathematicians used to commutative number systems. However, this mirrors the non-commutativity of geometric operations: rotating then reflecting differs from reflecting then rotating.

Notation: The notation $A^T$ for transpose was introduced later. Earlier texts used $A'$ or $\tilde{A}$ . In physics, $A^\dagger$ (dagger) denotes the conjugate transpose.

Computational Importance: Matrix multiplication is one of the most important operations in scientific computing. The naive $O(n^3)$ algorithm was long believed optimal until Strassen discovered an $O(n^{2.81})$ algorithm in 1969.

Modern Applications: Today, matrix operations underpin virtually all of machine learning (neural networks are sequences of matrix multiplications), computer graphics (transformations), physics simulations, and data science.

20. Geometric View of Matrix Operations

Multiplication as Transformation Composition

When you multiply matrices $AB$ , you're composing two geometric transformations:

First apply transformation $B$ (right matrix acts first)
Then apply transformation $A$

This explains non-commutativity: rotating 90° then reflecting horizontally gives a different result than reflecting first then rotating.

Column Picture

The columns of $A$ show where the standard basis vectors land:

A e_j = \text{column } j \text{ of } A

The entire transformation is determined by where it sends the basis vectors.

Transpose as Reflection

Geometrically, transpose reflects the matrix across its main diagonal. For symmetric matrices ( $A = A^T$ ), the matrix is unchanged by this reflection.

In inner product spaces, transpose relates to the adjoint: $\langle Ax, y \rangle = \langle x, A^T y \rangle$ .

What's Next?

Now that you understand matrix operations, the next topics extend these foundations:

Matrix Inverse: When does $A^{-1}$ exist and how to compute it
Elementary Matrices: Building blocks for Gaussian elimination
Matrix Rank: The fundamental dimension connecting all matrix concepts
Determinants: A scalar that captures key properties of square matrices

These topics complete our understanding of matrix algebra and prepare us for eigenvalue theory.

Matrix Operations Practice

12

Questions

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Accuracy

1

If

A

is

3 \times 4

and

B

is

4 \times 2

, what size is

AB

?

Easy

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2

Is matrix multiplication commutative?

Easy

Not attempted

3

(AB)^T = ?

Medium

Not attempted

4

If

A

is

m \times n

, what size is

A^T

?

Easy

Not attempted

5

(A + B)C = ?

Easy

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6

If

A

is symmetric, then

A = ?

Easy

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7

(ABC) = ?

Easy

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8

If

A

is

2 \times 3

and

B

is

2 \times 3

, is

AB

defined?

Easy

Not attempted

9

What is the

(i,j)

entry of

AB

?

Medium

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10

If

A \neq 0

and

B \neq 0

, is

AB \neq 0

?

Medium

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11

(A + B)^2 = ?

Hard

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12

Which matrices commute with ALL

n \times n

matrices?

Hard

Not attempted

Frequently Asked Questions

Why does matrix multiplication work the way it does?

It's designed so that $[S \circ T] = [S][T]$—the matrix of a composition is the product of matrices. The row-column dot product formula follows directly from how coordinates transform under composition of linear maps.

Why isn't matrix multiplication commutative?

Geometrically: rotate then reflect ≠ reflect then rotate. Algebraically: the composition of linear maps isn't commutative, and matrix multiplication represents composition. Even when both $AB$ and $BA$ are defined, they usually give different results.

What are block matrices good for?

They simplify large matrix computations and reveal structure. If matrices have compatible block partitions, you can multiply blocks like scalar entries (with care about order). Block diagonal matrices are particularly useful for decomposing problems.

What's the computational complexity of matrix multiplication?

Naive: $O(n^3)$. Best known (Coppersmith-Winograd variants): $O(n^{2.37...})$. In practice, optimized $O(n^3)$ algorithms (like Strassen's $O(n^{2.81})$) are often used due to lower constants.

When can I commute matrices?

Scalar matrices $\lambda I$ commute with everything. Diagonal matrices commute with each other. More generally, matrices commute if they share an eigenbasis—such matrices are simultaneously diagonalizable.

Why is $(AB)^T = B^T A^T$ (reversed order)?

Think of it as 'reversing the path.' Transposing swaps rows and columns. When you transpose a product, the roles of 'taking rows of A' and 'taking columns of B' get swapped, which reverses the multiplication order.

Can non-zero matrices multiply to give zero?

Yes! These are called zero divisors. For example, if $A$ has a non-trivial null space and $B$'s columns lie in that null space, then $AB = 0$ even though $A, B \neq 0$. This is fundamentally different from real number multiplication.

What is a matrix polynomial?

If $p(x) = a_n x^n + \cdots + a_1 x + a_0$, then $p(A) = a_n A^n + \cdots + a_1 A + a_0 I$. Matrix polynomials are central to the Cayley-Hamilton theorem and matrix functions.

What's the identity matrix's role?

The identity matrix $I$ satisfies $AI = IA = A$ for any compatible matrix $A$. It represents the identity linear map. In matrix multiplication, it plays the role that 1 plays in number multiplication.

How do I multiply matrices by hand efficiently?

For each entry $(i,j)$ of the result, compute the dot product of row $i$ of the first matrix with column $j$ of the second. Organize your work systematically and double-check dimensions first.