LA-5.2

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Properties of Determinants

Determinants possess remarkable algebraic properties that make them indispensable tools for matrix analysis. These properties—multiplicativity, behavior under transpose, and effects of elementary operations—form the computational backbone of linear algebra.

2-3 hours Core Level 10 Objectives

Learning Objectives

Prove and apply the multiplicativity property: det(AB) = det(A)det(B)
Master the transpose property: det(A^T) = det(A) and its consequences
Understand the deep connection between determinant and invertibility
Compute det(A^{-1}) = 1/det(A) for invertible matrices
Apply row and column operation effects on determinants systematically
Calculate determinants of all three types of elementary matrices
Master determinants of triangular matrices (upper, lower, diagonal)
Compute determinants of block triangular and block diagonal matrices
Understand det(kA) = k^n det(A) for n×n matrices
Recognize and avoid common mistakes involving determinant properties

Prerequisites

Determinant definition via axioms, permutations, and cofactors (LA-5.1)
Matrix multiplication and the associative law
Elementary matrices and their relationship to row operations
Matrix inverse and the equation AA⁻¹ = I
Basic properties of transpose
Understanding of linear independence and rank

1. The Multiplicativity Property

The most fundamental property of determinants is multiplicativity: the determinant of a product equals the product of determinants. This connects matrix algebra to scalar algebra and has geometric meaning—composing transformations multiplies volume scaling factors.

Theorem 5.4: Multiplicativity of Determinant

For any $n \times n$ matrices $A$ and $B$ over a field $F$ :

\det(AB) = \det(A) \cdot \det(B)

Proof:

Method 1 (Axiomatic): Fix matrix $A$ and define $f(B) = \det(AB)$ .

We verify $f$ satisfies the three axioms:

Multilinearity: $f$ is multilinear in rows of $B$ (since $AB$ has rows that are linear in $B$ 's rows)
Alternating: Swapping rows of $B$ swaps corresponding rows of $AB$
Normalization: $f(I) = \det(AI) = \det(A)$

By uniqueness, $f(B) = \det(A) \cdot \det(B)$ .

Method 2 (Elementary matrices): If $B$ is invertible, write $B = E_1 \cdots E_k$ . Since $\det(AE_i) = \det(A) \cdot \det(E_i)$ for each elementary matrix, by induction $\det(AB) = \det(A) \cdot \det(B)$ .

If $B$ is not invertible, then $AB$ is also not invertible, so both sides equal 0.

∎

Example 5.10: Verifying Multiplicativity

Let $A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix}$ .

\det(A) = 6-1 = 5, \quad \det(B) = 4-0 = 4

AB = \begin{pmatrix} 2 & 8 \\ 1 & 14 \end{pmatrix}, \quad \det(AB) = 28-8 = 20 = 5 \cdot 4 \checkmark

Corollary 5.2: Power of Determinant

For any $n \times n$ matrix $A$ and positive integer $k$ :

\det(A^k) = (\det A)^k

Proof:

By induction using multiplicativity: $\det(A^k) = \det(A^{k-1}) \cdot \det(A) = (\det A)^{k-1} \cdot \det(A) = (\det A)^k$ .

∎

Remark 5.1: Geometric Interpretation

If $T_A$ and $T_B$ are linear transformations with matrices $A$ and $B$ :

$|\det(A)|$ = volume scaling factor of $T_A$
$|\det(AB)| = |\det(A)| \cdot |\det(B)|$ = volume scaling of $T_A \circ T_B$

2. Determinant of Transpose

The transpose exchanges rows and columns. Remarkably, this preserves the determinant, establishing a powerful duality: every row property has a column counterpart.

Theorem 5.5: Transpose Property

For any $n \times n$ matrix $A$ :

\det(A^T) = \det(A)

Proof:

Using permutation formula: We have $(A^T)_{ij} = a_{ji}$ , so:

\det(A^T) = \sum_{\sigma \in S_n} (-1)^{\tau(\sigma)} a_{\sigma(1),1} \cdots a_{\sigma(n),n}

The map $\sigma \mapsto \sigma^{-1}$ is a bijection on $S_n$ with $\tau(\sigma) = \tau(\sigma^{-1})$ .

The products are identical (just reordered), so the sums are equal.

∎

Remark 5.2: Row-Column Duality

Since $\det(A^T) = \det(A)$ , all row properties have column versions:

Swapping two columns negates det
Scaling a column by $c$ multiplies det by $c$
Adding a multiple of one column to another preserves det
Two identical columns means det = 0

Example 5.11: Transpose Verification

$A = \begin{pmatrix} 1 & 2 \\ 0 & 4 \end{pmatrix}$ (upper triangular), $\det(A) = 4$ .

$A^T = \begin{pmatrix} 1 & 0 \\ 2 & 4 \end{pmatrix}$ (lower triangular), $\det(A^T) = 4 \checkmark$

Corollary 5.3: Products with Transpose

\det(A \cdot A^T) = \det(A^T \cdot A) = (\det A)^2

3. Determinant and Invertibility

The connection between determinants and invertibility is fundamental. A matrix is invertible if and only if its determinant is nonzero—a simple scalar test for an otherwise complex property.

Theorem 5.6: Determinant of Inverse

If $A$ is an invertible $n \times n$ matrix:

\det(A^{-1}) = \frac{1}{\det(A)} = (\det A)^{-1}

Proof:

Apply multiplicativity to $A \cdot A^{-1} = I$ :

\det(A) \cdot \det(A^{-1}) = \det(I) = 1

Since $A$ is invertible, $\det(A) \neq 0$ , so $\det(A^{-1}) = 1/\det(A)$ .

∎

Theorem 5.7: Invertibility Criterion

For any $n \times n$ matrix $A$ :

A \text{ is invertible} \iff \det(A) \neq 0

Proof:

(⟹) If $A$ is invertible, write $A = E_1 E_2 \cdots E_k$ . Each $\det(E_i) \neq 0$ , so $\det(A) \neq 0$ .

(⟸) If $A$ is not invertible, columns are dependent. By multilinearity, any column can be written as a linear combination of others, giving det = 0.

∎

Example 5.12: Checking Invertibility

Is $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$ invertible?

\det(A) = 1(45-48) - 2(36-42) + 3(32-35) = -3 + 12 - 9 = 0

Since $\det(A) = 0$ , $A$ is not invertible.

Remark 5.3: Equivalent Conditions

For $n \times n$ matrix $A$ , these are equivalent:

$A$ is invertible
$\det(A) \neq 0$
$\text{rank}(A) = n$
Columns (or rows) are linearly independent
$Ax = 0$ has only the trivial solution

4. Effects of Row and Column Operations

Understanding how elementary operations affect determinants is crucial for computation. These effects follow from the axiomatic definition and enable efficient algorithms.

Theorem 5.8: Effects of Row Operations

Let $A$ be $n \times n$ and $A'$ result from a row operation:

Operation	Notation	Effect
Swap rows $i$ , $j$	$R_i \leftrightarrow R_j$	$\det(A') = -\det(A)$
Scale row $i$ by $c$	$R_i \to c \cdot R_i$	$\det(A') = c \cdot \det(A)$
Add $k$ × row $j$ to row $i$	$R_i \to R_i + kR_j$	$\det(A') = \det(A)$

Proof:

Type I (Swap): Follows from alternating property.

Type II (Scale): By multilinearity: $\det(\ldots, c\alpha_i, \ldots) = c \cdot \det(\ldots, \alpha_i, \ldots)$ .

Type III (Add): $\det(\ldots, \alpha_i + k\alpha_j, \ldots, \alpha_j, \ldots) = \det(A) + k \cdot 0 = \det(A)$ .

∎

Example 5.13: Row Reduction to Compute Determinant

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

Step 1: Factor 2 from row 1: $= 2 \cdot \det\begin{pmatrix} 1 & 2 & 3 \\ 1 & 2 & 5 \\ 3 & 1 & 4 \end{pmatrix}$

Step 2: $R_2 - R_1, R_3 - 3R_1$ : $= 2 \cdot \det\begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 2 \\ 0 & -5 & -5 \end{pmatrix}$

Step 3: $R_2 \leftrightarrow R_3$ : $= -2 \cdot \det\begin{pmatrix} 1 & 2 & 3 \\ 0 & -5 & -5 \\ 0 & 0 & 2 \end{pmatrix}$

Step 4: Upper triangular: $= -2 \cdot 1 \cdot (-5) \cdot 2 = 20$

Remark 5.4: Column Operations

By transpose property, column operations have identical effects on determinant.

5. Determinants of Elementary Matrices

Elementary matrices are obtained by performing one row operation on the identity. Their determinants follow directly from the effect on det(I) = 1.

Theorem 5.9: Determinants of Elementary Matrices

The three types have determinants:

Row swap: $\det(E_{ij}) = -1$
Row scaling: $\det(E_i(c)) = c$
Row addition: $\det(E_{ij}(k)) = 1$

Example 5.14: Elementary Matrix Determinants

E_{12} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}

$\det = -1$

E_1(3) = \begin{pmatrix} 3 & 0 \\ 0 & 1 \end{pmatrix}

$\det = 3$

E_{12}(2) = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}

$\det = 1$

Corollary 5.4: Left Multiplication Effect

$\det(E \cdot A) = \det(E) \cdot \det(A)$ , confirming row operation effects.

Remark 5.6: Right Multiplication (Column Operations)

Right multiplication $A \cdot E$ performs column operations:

$A \cdot E_{ij}$ swaps columns $i$ and $j$
$A \cdot E_i(c)$ scales column $i$ by $c$
$A \cdot E_{ij}(k)$ adds $k$ times column $i$ to column $j$

Same determinant effects: $\det(A \cdot E) = \det(A) \cdot \det(E)$ .

Example 5.18: Factoring a Matrix via Elementary Matrices

If $A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}$ , we can row reduce:

$R_1 \leftrightarrow R_2$ : $E_{12} \cdot A = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$

$R_2 - 2R_1$ : $E_{21}(-2) \cdot E_{12} \cdot A = \begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix}$

So $\det(A) = \det(E_{21}(-2))^{-1} \cdot \det(E_{12})^{-1} \cdot 1 \cdot (-1) = 1 \cdot (-1) \cdot (-1) = 1$

Verify: $\det(A) = 2 \cdot 2 - 3 \cdot 1 = 1 \checkmark$

6. Scalar Multiplication

Theorem 5.10: Scalar Multiple Property

For an $n \times n$ matrix $A$ and scalar $k$ :

\det(kA) = k^n \det(A)

Proof:

Writing $A$ with rows $\alpha_1, \ldots, \alpha_n$ :

\det(kA) = \det(k\alpha_1, \ldots, k\alpha_n) = k^n \det(A)

Each of $n$ rows contributes a factor of $k$ by multilinearity.

∎

Example 5.15: Scalar Multiplication

If $A$ is $4 \times 4$ with $\det(A) = 3$ :

\det(2A) = 2^4 \cdot 3 = 48, \quad \det(-A) = (-1)^4 \cdot 3 = 3

Remark 5.5: Common Error

Warning: $\det(kA) \neq k \cdot \det(A)$ unless $n = 1$ !

Example 5.19: Sign of det(-A)

For an $n \times n$ matrix:

\det(-A) = (-1)^n \det(A)

$n$ even: $\det(-A) = \det(A)$
$n$ odd: $\det(-A) = -\det(A)$

Example: For 3×3 matrix with det(A) = 5: $\det(-A) = (-1)^3 \cdot 5 = -5$

Corollary 5.5: Determinant of Similar Matrices

If $B = P^{-1}AP$ (similar matrices), then:

\det(B) = \det(P^{-1}AP) = \det(P^{-1}) \det(A) \det(P) = \frac{1}{\det(P)} \det(A) \det(P) = \det(A)

Conclusion: Similar matrices have the same determinant.

7. Determinants of Special Matrices

Theorem 5.11: Triangular Matrices

For upper/lower triangular or diagonal matrices:

\det(A) = a_{11} \cdot a_{22} \cdots a_{nn} = \prod_{i=1}^{n} a_{ii}

Example 5.16: Triangular Determinants

\det\begin{pmatrix} 2 & 3 & 7 \\ 0 & -1 & 5 \\ 0 & 0 & 4 \end{pmatrix} = 2 \cdot (-1) \cdot 4 = -8

Theorem 5.12: Block Triangular Matrices

For block triangular (A is k×k, B is r×r):

\det\begin{pmatrix} A & C \\ O & B \end{pmatrix} = \det(A) \cdot \det(B)

\det\begin{pmatrix} A & O \\ C & B \end{pmatrix} = \det(A) \cdot \det(B)

Proof:

Expand along the first column/row repeatedly. The block of zeros means only one cofactor is nonzero in each expansion step.

∎

Example 5.17: Block Triangular

$\det\begin{pmatrix} 1 & 2 & 3 & 4 \\ 0 & 5 & 6 & 7 \\ 0 & 0 & 8 & 9 \\ 0 & 0 & 1 & 2 \end{pmatrix}$

With $A = \begin{pmatrix} 1 & 2 \\ 0 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 8 & 9 \\ 1 & 2 \end{pmatrix}$ :

\det = 5 \cdot (16-9) = 5 \cdot 7 = 35

Theorem 5.13: Anti-Block Diagonal

\det\begin{pmatrix} O & A \\ B & O \end{pmatrix} = (-1)^{kr} \det(A) \cdot \det(B)

where A is k×r and B is r×k.

Definition 5.5: Schur Complement

For block matrix $M = \begin{pmatrix} A & B \\ C & D \end{pmatrix}$ with $A$ invertible:

M/A = D - CA^{-1}B

is called the Schur complement of $A$ in $M$ .

Theorem 5.14: Determinant via Schur Complement

If $A$ is invertible:

\det\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det(A) \cdot \det(D - CA^{-1}B) = \det(A) \cdot \det(M/A)

Proof:

Block LU decomposition (Gaussian block elimination):

\begin{pmatrix} I & O \\ -CA^{-1} & I \end{pmatrix} \begin{pmatrix} A & B \\ C & D \end{pmatrix} = \begin{pmatrix} A & B \\ O & D - CA^{-1}B \end{pmatrix}

The left matrix has det = 1 (block lower triangular with I on diagonal), so:

\det\begin{pmatrix} A & B \\ C & D \end{pmatrix} = \det\begin{pmatrix} A & B \\ O & D-CA^{-1}B \end{pmatrix} = \det(A) \cdot \det(D - CA^{-1}B)

∎

Example 5.20: Schur Complement Calculation

Compute $\det\begin{pmatrix} 2 & 1 & 0 \\ 1 & 3 & 1 \\ 0 & 1 & 2 \end{pmatrix}$ using Schur complement.

Take $A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$ , $B = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ , $C = (0, 1)$ , $D = 2$ .

A^{-1} = \frac{1}{5}\begin{pmatrix} 3 & -1 \\ -1 & 2 \end{pmatrix}, \quad CA^{-1}B = \frac{1}{5}(0,1)\begin{pmatrix} 3 & -1 \\ -1 & 2 \end{pmatrix}\begin{pmatrix} 0 \\ 1 \end{pmatrix} = \frac{2}{5}

\det = \det(A) \cdot (D - CA^{-1}B) = 5 \cdot (2 - \frac{2}{5}) = 5 \cdot \frac{8}{5} = 8

Remark 5.7: Vandermonde Determinant

A classic special determinant is the Vandermonde determinant:

\det\begin{pmatrix} 1 & x_1 & x_1^2 & \cdots & x_1^{n-1} \\ 1 & x_2 & x_2^2 & \cdots & x_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & x_n & x_n^2 & \cdots & x_n^{n-1} \end{pmatrix} = \prod_{1 \leq i < j \leq n} (x_j - x_i)

This is nonzero iff all $x_i$ are distinct, which is key in polynomial interpolation.

8. Common Mistakes to Avoid

Determinants have counterintuitive properties. Here are critical errors to avoid.

Mistake 1: det(A+B) = det(A) + det(B)

Wrong! Determinant is NOT additive.

Example: A = B = I₂. det(A) = det(B) = 1, but det(A+B) = det(2I) = 4 ≠ 2.

Mistake 2: det(cA) = c·det(A)

Wrong! It's $c^n \det(A)$ for n×n matrices.

Example: det(2I₃) = 2³ = 8, not 2.

Mistake 3: Forgetting Sign in Row Swaps

Each row swap multiplies det by -1.

Tip: Count swaps during row reduction and adjust sign at the end.

Mistake 4: Confusing Row vs Matrix Scaling

Scaling ONE row by c → multiply det by c

Scaling WHOLE matrix by c → multiply det by c^n

Mistake 5: det(AB) = det(BA)?

True! This one is actually correct: det(AB) = det(A)det(B) = det(B)det(A) = det(BA).

Note: Even though AB ≠ BA in general, their determinants are always equal.

9. Key Takeaways

Multiplicative

det(AB) = det(A)det(B)

Transpose

det(A^T) = det(A)

Invertibility

A invertible ⟺ det(A) ≠ 0

Scalar Multiple

det(kA) = k^n det(A)

10. Practice Problems

Problem 1

If $\det(A) = 3$ and $\det(B) = -2$ , find $\det(A^2 B^{-1})$ .

Problem 2

Let $A$ be 3×3 with $\det(A) = 4$ . Find $\det(2A^T)$ and $\det(A^{-1})$ .

Problem 3

Compute $\det\begin{pmatrix} 3 & 0 & 0 & 0 \\ 1 & 2 & 0 & 0 \\ 4 & 5 & -1 & 0 \\ 2 & 3 & 6 & 4 \end{pmatrix}$ .

Problem 4

Prove: If $A^2 = A$ (idempotent), then $\det(A) \in \{0, 1\}$ .

Problem 5

If $A$ is orthogonal ( $A^T A = I$ ), prove $\det(A) = \pm 1$ .

Problem 6

Compute $\det\begin{pmatrix} 1+a & 1 & 1 \\ 1 & 1+b & 1 \\ 1 & 1 & 1+c \end{pmatrix}$ .

Solutions

Solution 1

\det(A^2 B^{-1}) = \det(A)^2 \cdot \det(B)^{-1} = 9 \cdot (-\frac{1}{2}) = -\frac{9}{2}

Solution 2

\det(2A^T) = 2^3 \cdot \det(A^T) = 8 \cdot 4 = 32

\det(A^{-1}) = \frac{1}{\det(A)} = \frac{1}{4}

Solution 3

Lower triangular, so det = product of diagonal:

\det = 3 \cdot 2 \cdot (-1) \cdot 4 = -24

Solution 4

From $A^2 = A$ : $\det(A^2) = \det(A)$

So $(\det A)^2 = \det A$ , meaning $\det A (\det A - 1) = 0$ .

Therefore $\det A = 0$ or $\det A = 1$ .

Solution 5

From $A^T A = I$ :

\det(A^T A) = \det(I) = 1

\det(A^T) \cdot \det(A) = (\det A)^2 = 1

Therefore $\det A = \pm 1$ .

Solution 6

Use row/column operations. Subtract row 1 from rows 2, 3:

\det = \det\begin{pmatrix} 1+a & 1 & 1 \\ -a & b & 0 \\ -a & 0 & c \end{pmatrix}

Expand along column 3:

= 1 \cdot \det\begin{pmatrix} -a & b \\ -a & 0 \end{pmatrix} + c \cdot \det\begin{pmatrix} 1+a & 1 \\ -a & b \end{pmatrix}

= 1 \cdot (0 - (-ab)) + c \cdot ((1+a)b - (-a)) = ab + c(b + ab + a)

= ab + bc + abc + ac = ab + bc + ca + abc

11. Quick Reference Summary

Core Properties

Property	Formula
Multiplicativity	$\det(AB) = \det(A) \det(B)$
Transpose	$\det(A^T) = \det(A)$
Inverse	$\det(A^{-1}) = 1/\det(A)$
Power	$\det(A^k) = (\det A)^k$
Scalar multiple	$\det(kA) = k^n \det(A)$
Invertibility	$A$ invertible $\iff \det(A) \neq 0$

Elementary Matrix Determinants

Type	Matrix	Determinant
Row swap	$E_{ij}$	$-1$
Row scale	$E_i(c)$	$c$
Row addition	$E_{ij}(k)$	$1$

Special Matrix Determinants

Type	Formula
Triangular	Product of diagonal: $a_{11} \cdots a_{nn}$
Block triangular	$\det(A) \cdot \det(B)$
Anti-block diagonal	$(-1)^{kr} \det(A) \det(B)$
Schur complement	$\det(A) \cdot \det(D - CA^{-1}B)$
Similar matrices	$\det(P^{-1}AP) = \det(A)$

Row Operation Effects

Operation	Effect on det
Swap two rows	Multiply by $-1$
Scale row by $c$	Multiply by $c$
Add multiple of row	No change

12. Historical Notes

Cauchy (1812): Proved the multiplicative property det(AB) = det(A)det(B), establishing determinants as a proper algebraic theory beyond just computational formulas. His work laid the foundation for understanding determinants as algebraic objects with consistent rules.

Jacobi (1841): Systematically developed properties of determinants and introduced the Jacobian determinant, connecting linear algebra to multivariable calculus. The Jacobian measures how volume changes under coordinate transformations.

Sylvester (1851): Coined the term "matrix" and developed the theory of matrix determinants alongside Arthur Cayley. Introduced the concept of minors and contributed to the understanding of block matrices.

Weierstrass: Contributed to the axiomatic understanding, showing that the three axioms (multilinear, alternating, normalized) uniquely determine the determinant. This abstract approach became foundational for modern linear algebra.

Modern view: Determinant is now understood as the unique alternating multilinear form on n vectors that equals 1 on the standard basis—connecting to exterior algebra and differential forms. In this view, det is a volume form.

13. Applications Preview

Determinant properties are used throughout mathematics and applications. Here are key areas where these properties are essential.

Cramer's Rule

Solve $Ax = b$ using determinants: $x_i = \det(A_i)/\det(A)$ . Requires det(A) ≠ 0 (invertibility).

Eigenvalue Equations

Eigenvalues satisfy $\det(A - \lambda I) = 0$ . The characteristic polynomial uses det properties extensively.

Change of Variables

In multivariable integrals: $dV' = |\det J| \, dV$ where $J$ is the Jacobian matrix.

Linear Independence

Vectors $v_1, \ldots, v_n$ are linearly independent iff $\det[v_1 \cdots v_n] \neq 0$ .

Area and Volume

Area of parallelogram: $|\det[v, w]|$ . Volume of parallelepiped: $|\det[u, v, w]|$ .

Orientation

Sign of det determines orientation. Positive = preserves handedness, negative = reverses (like reflection).

What's Next?

Now that you've mastered determinant properties, the next topics explore:

Computation Methods: Efficient techniques including row reduction and cofactor tricks
Laplace Expansion: Expansion along any row or column, generalized
Adjugate Matrix: The transpose of the cofactor matrix and Cramer's Rule

Key Skills You've Learned

Apply multiplicativity: det(AB) = det(A)det(B)
Use transpose property for row-column duality
Check invertibility via determinant
Track determinant changes through row operations
Compute determinants of triangular and block matrices
Use Schur complement for general block matrices

Chapter Summary

This module covered the fundamental algebraic properties of determinants. The key insight is that determinant behaves like a multiplicative homomorphism from matrices to scalars: it respects products but not sums. Combined with the transpose property, this gives determinants their computational power.

Theorems Covered

Practice Questions

FAQs Answered

Determinant Properties Practice

Questions

Correct

Accuracy

\det(AB) =

Easy

Not attempted

\det(A^T) =

Easy

Not attempted

A

is invertible,

\det(A^{-1}) =

Medium

Not attempted

Determinant of an upper triangular matrix equals:

Easy

Not attempted

\det(cA)

for an

n \times n

matrix

A

equals:

Medium

Not attempted

A

has two identical rows, then:

Easy

Not attempted

Adding a multiple of row

j

to row

i

(with

i \neq j

Easy

Not attempted

\det(A^k) =

Medium

Not attempted

The determinant of the elementary matrix

E_{ij}

(row swap) is:

Easy

Not attempted

The determinant of

E_i(c)

(scale row

i

c

) is:

Easy

Not attempted

The determinant of

E_{ij}(k)

(add

k

times row

j

to row

i

) is:

Easy

Not attempted

For block diagonal matrix

\begin{pmatrix} A & O \\ O & B \end{pmatrix}

\det =

Medium

Not attempted

A

3 \times 3

and

\det(A) = 2

, then

\det(2A) =

Medium

Not attempted

\det(A) = 0

, then:

Easy

Not attempted

For any square matrix

A

\det(A \cdot A^T) =

Medium

Not attempted

Frequently Asked Questions

Why is det(AB) = det(A)det(B)?

The function f(B) = det(AB) is multilinear and alternating in the rows of B, with f(I) = det(A). By the uniqueness of determinants, f(B) must equal det(A)·det(B). Geometrically, composing linear transformations multiplies their volume scaling factors.

How do row operations affect the determinant?

There are three types: (1) Swap two rows → multiply det by -1; (2) Scale a row by c → multiply det by c; (3) Add a multiple of one row to another → det unchanged. These follow directly from the axiomatic definition (alternating, multilinear, normalized).

What's special about triangular matrix determinants?

For both upper and lower triangular matrices, det = product of diagonal entries. This makes Gaussian elimination extremely efficient: reduce to triangular form (tracking sign changes from swaps), then multiply the diagonal. This is O(n³) vs O(n!) for direct computation.

How does det relate to invertibility?

A square matrix A is invertible if and only if det(A) ≠ 0. Geometrically, det = 0 means the transformation collapses space (zero volume). Algebraically, det = 0 means the columns are linearly dependent, so A has a non-trivial kernel and cannot be injective.

Does det(A+B) = det(A) + det(B)?

NO! This is a common mistake. Determinant is NOT additive. It's only linear in each row separately (multilinear). Counterexample: A = B = I₂ gives det(A) = det(B) = 1, but det(A+B) = det(2I) = 4 ≠ 2.

Why does det(A^T) = det(A)?

In the permutation formula, transposing swaps row and column indices. Each permutation σ appears paired with its inverse σ⁻¹ (which has the same sign), so the sum is unchanged. This duality means all row properties have column versions.

How do I compute det of a block matrix?

For block triangular matrices: det([A,B;O,D]) = det(A)·det(D) and det([A,O;C,B]) = det(A)·det(B). For general block matrices [A,B;C,D], if A is invertible: det = det(A)·det(D - CA⁻¹B). The quantity D - CA⁻¹B is called the Schur complement.

What are the determinants of elementary matrices?

det(E_ij) = -1 (row swap), det(E_i(c)) = c (scale row by c), det(E_ij(k)) = 1 (add k times row j to row i). These follow from applying each operation to the identity matrix and using the determinant axioms.

Why is det(kA) = k^n·det(A) and not k·det(A)?

When you multiply a matrix by scalar k, EVERY row gets multiplied by k. Since det is linear in each row, multiplying n rows by k gives k^n factor. For example, det(2·I₃) = 2³ = 8, not 2.

Can determinant be negative?

Yes! The sign of det indicates orientation. det > 0 means the transformation preserves orientation, det < 0 means it reverses orientation (like a reflection). Only |det| measures volume scaling.

How is det(A⁻¹) related to det(A)?

For invertible A: det(A⁻¹) = 1/det(A). This follows from det(A)·det(A⁻¹) = det(A·A⁻¹) = det(I) = 1. Note that A must be invertible (det(A) ≠ 0) for this to make sense.

What happens to det when I multiply a single row by a scalar?

Multiplying ONE row by scalar c multiplies det by c (multilinearity). This is different from multiplying the WHOLE matrix by c, which gives c^n·det(A). Common mistake: confusing these two operations.