LA-4.3

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Matrix Inverse

The inverse of a matrix undoes its action. Invertible matrices correspond to bijective linear maps.

Learning Objectives

Define the inverse of a square matrix
Prove uniqueness of the inverse
Apply the fundamental characterizations of invertibility
Compute inverses using Gaussian elimination
Apply properties of inverse: product, transpose, powers
Understand the connection between invertible matrices and isomorphisms
Recognize when a matrix is NOT invertible
Use the inverse to solve systems of equations

Prerequisites

Matrix operations (LA-4.2)
Matrix representation (LA-4.1)
Linear map composition
Gaussian elimination basics

1. Definition and Motivation

Just as a bijective function has an inverse that "undoes" it, an invertible linear map has an inverse map. The matrix inverse is the matrix representation of this inverse map.

Definition 4.13: Inverse Matrix

Let $A \in M_n(F)$ be an $n \times n$ matrix. If there exists a matrix $B \in M_n(F)$ such that:

AB = BA = I_n

then $A$ is called invertible (or nonsingular), and $B$ is called the inverse of $A$ , denoted $A^{-1}$ .

Remark 4.12: Singular vs Nonsingular

A square matrix that is NOT invertible is called singular. The terminology comes from the fact that singular matrices represent "exceptional" or "degenerate" cases.

Theorem 4.18: Uniqueness of the Inverse

If $A$ is invertible, then its inverse $A^{-1}$ is unique.

Proof:

Suppose $B$ and $C$ both satisfy $AB = BA = I$ and $AC = CA = I$ . Then:

B = BI = B(AC) = (BA)C = IC = C

Therefore $B = C$ , proving uniqueness.

∎

Remark 4.13: Connection to Isomorphisms

A matrix $A$ is invertible if and only if the linear map it represents is an isomorphism (bijective). The inverse matrix represents the inverse map.

2. One-Sided Inverses for Square Matrices

Theorem 4.19: One-Sided Implies Two-Sided

For square matrices $A, B \in M_n(F)$ :

AB = I \iff BA = I \iff A, B \text{ are both invertible}

Proof:

Suppose $AB = I$ . We show $B$ is invertible with inverse $A$ :

Step 1: Show columns of $B$ are linearly independent.

If $Bx = 0$ , then $x = Ix = (AB)x = A(Bx) = A \cdot 0 = 0$ .

Step 2: Since $B$ has independent columns, it's invertible.

Step 3: From $AB = I$ , right-multiply by $B^{-1}$ : $A = B^{-1}$ .

Therefore $BA = BB^{-1} = I$ .

∎

Remark 4.14: Warning: Non-Square Matrices

This fails for non-square matrices! A left inverse doesn't imply a right inverse:

A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}

Then $BA = I_2$ but $AB \neq I_3$ .

3. Characterizations of Invertibility

Theorem 4.20: Invertibility Equivalences (The Invertible Matrix Theorem)

For an $n \times n$ matrix $A$ , the following are equivalent (TFAE):

$A$ is invertible
$\det(A) \neq 0$
$\text{rank}(A) = n$
The columns of $A$ are linearly independent
The rows of $A$ are linearly independent
$Ax = 0$ has only the trivial solution $x = 0$
$Ax = b$ has a unique solution for every $b$
$A$ is a product of elementary matrices
$A$ is row equivalent to $I_n$
The reduced row echelon form of $A$ is $I_n$
$\ker(A) = \{0\}$ (trivial null space)
$\text{im}(A) = F^n$ (columns span all of $F^n$ )

Remark 4.15: Power of the Theorem

This theorem is central to linear algebra. It connects seemingly different concepts: invertibility, determinants, rank, independence, solvability of equations, and elementary matrices.

4. Properties of the Inverse

Theorem 4.21: Inverse Properties

For invertible matrices $A$ , $B$ and scalar $\lambda \neq 0$ :

$(A^{-1})^{-1} = A$
$(AB)^{-1} = B^{-1}A^{-1}$ (order reverses!)
$(A^n)^{-1} = (A^{-1})^n$
$(\lambda A)^{-1} = \lambda^{-1} A^{-1}$
$(A^T)^{-1} = (A^{-1})^T$
$\det(A^{-1}) = 1/\det(A)$

Proof:

(2): Verify directly:

(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I

By uniqueness, $(AB)^{-1} = B^{-1}A^{-1}$ .

(5): From $AA^{-1} = I$ , take transpose of both sides:

(A^{-1})^T A^T = I^T = I

This shows $(A^{-1})^T$ is a left inverse of $A^T$ . For square matrices, this means $(A^T)^{-1} = (A^{-1})^T$ .

∎

Corollary 4.7: Product of Inverses

For invertible matrices $A_1, A_2, \ldots, A_k$ :

(A_1 A_2 \cdots A_k)^{-1} = A_k^{-1} \cdots A_2^{-1} A_1^{-1}

Remark 4.16: Cancellation Law

If $A$ is invertible and $AB = AC$ , then $B = C$ . Multiply both sides by $A^{-1}$ on the left.

5. Computing the Inverse

Theorem 4.22: Gaussian Elimination Method

To compute $A^{-1}$ :

Form the augmented matrix $[A | I]$
Row reduce to reduced echelon form
If the left part becomes $I$ , the right part is $A^{-1}$
If $A$ cannot be reduced to $I$ , then $A$ is not invertible

Example 4.8: Computing an Inverse

Find the inverse of $A = \begin{pmatrix} 1 & -1 & 1 \\ 0 & 1 & 2 \\ 1 & 0 & 4 \end{pmatrix}$ .

Solution: Form $[A | I]$ and row reduce:

\begin{pmatrix} 1 & -1 & 1 & | & 1 & 0 & 0 \\ 0 & 1 & 2 & | & 0 & 1 & 0 \\ 1 & 0 & 4 & | & 0 & 0 & 1 \end{pmatrix}

After row operations: $R_3 \to R_3 - R_1$ , then eliminate to get:

\begin{pmatrix} 1 & 0 & 0 & | & 4 & 4 & -3 \\ 0 & 1 & 0 & | & 2 & 3 & -2 \\ 0 & 0 & 1 & | & -1 & -1 & 1 \end{pmatrix}

Therefore:

A^{-1} = \begin{pmatrix} 4 & 4 & -3 \\ 2 & 3 & -2 \\ -1 & -1 & 1 \end{pmatrix}

Remark 4.17: Why This Works

Row reducing $[A|I]$ applies the same sequence of elementary row operations to both $A$ and $I$ . If $A$ reduces to $I$ , then $I$ becomes $A^{-1}$ because elementary operations are equivalent to left-multiplying by elementary matrices.

6. Solving Systems with Inverses

Theorem 4.23: Inverse Solution Formula

If $A$ is invertible, the unique solution to $Ax = b$ is:

x = A^{-1}b

Proof:

Multiply both sides of $Ax = b$ by $A^{-1}$ on the left:

A^{-1}(Ax) = A^{-1}b \implies (A^{-1}A)x = A^{-1}b \implies Ix = A^{-1}b \implies x = A^{-1}b

∎

Remark 4.18: Practical Considerations

While the formula $x = A^{-1}b$ is elegant, in practice:

Computing $A^{-1}$ explicitly is often unnecessary
Direct solution (e.g., LU decomposition) is usually faster and more stable
$A^{-1}$ is useful when solving $Ax = b$ for many different $b$ vectors

7. Worked Examples

Example 1: 2×2 Inverse Formula

For $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with $ad - bc \neq 0$ :

A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}

Example: $A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$

A^{-1} = \frac{1}{6-5} \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}

Example 2: Diagonal Matrix Inverse

If $D = \text{diag}(d_1, \ldots, d_n)$ with all $d_i \neq 0$ :

D^{-1} = \text{diag}(1/d_1, \ldots, 1/d_n)

Each diagonal entry is replaced by its reciprocal.

Example 3: Inverse of Product

Problem: If $A^{-1} = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B^{-1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ , find $(AB)^{-1}$ .

Solution:

(AB)^{-1} = B^{-1}A^{-1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} = \begin{pmatrix} 3 & 4 \\ 1 & 2 \end{pmatrix}

Example 4: When is a Matrix Singular?

Problem: For which values of $k$ is $A = \begin{pmatrix} 1 & k \\ k & 4 \end{pmatrix}$ singular?

Solution: $A$ is singular iff $\det(A) = 0$ :

\det(A) = 4 - k^2 = 0 \implies k = \pm 2

$A$ is singular when $k = 2$ or $k = -2$ .

8. Common Mistakes

Mistake 1: Wrong Order for Product Inverse

$(AB)^{-1} \neq A^{-1}B^{-1}$ . The correct formula is $(AB)^{-1} = B^{-1}A^{-1}$ . Remember: order reverses, just like for transpose.

Mistake 2: Assuming Non-Square Matrices Have Inverses

Only square matrices can have (two-sided) inverses. An $m \times n$ matrix with $m \neq n$ cannot be invertible, though it may have one-sided inverses or pseudoinverses.

Mistake 3: Assuming Every Square Matrix is Invertible

Many square matrices are singular (not invertible). Check the determinant, rank, or column independence before assuming invertibility.

Mistake 4: Inverse of Sum

$(A + B)^{-1} \neq A^{-1} + B^{-1}$ . There is no simple formula for the inverse of a sum.

9. Key Takeaways

Definition

$AA^{-1} = A^{-1}A = I$ . The inverse undoes the matrix.

Invertibility Test

$\det(A) \neq 0$ iff $A$ invertible iff $\text{rank}(A) = n$ .

Product Inverse

$(AB)^{-1} = B^{-1}A^{-1}$ . Order reverses!

Computation

Row reduce $[A|I]$ to $[I|A^{-1}]$ .

10. Quick Reference Summary

Property	Formula
Definition	$AA^{-1} = A^{-1}A = I$
Double Inverse	$(A^{-1})^{-1} = A$
Product Inverse	$(AB)^{-1} = B^{-1}A^{-1}$
Transpose-Inverse	$(A^T)^{-1} = (A^{-1})^T$
Scalar Inverse	$(\lambda A)^{-1} = \lambda^{-1}A^{-1}$
Determinant	$\det(A^{-1}) = 1/\det(A)$

11. Additional Practice Problems

Problem 1

Compute the inverse of $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}$ using the 2×2 formula.

Problem 2

Prove that if $A^2 = I$ (involutory matrix), then $A = A^{-1}$ .

Problem 3

If $A$ is invertible, show that $(A^k)^{-1} = (A^{-1})^k$ for all positive integers $k$ .

Problem 4

Find the inverse of $\begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 0 \end{pmatrix}$ using row reduction.

Problem 5

Prove that if $A$ and $B$ are invertible and commute ( $AB = BA$ ), then $A^{-1}$ and $B^{-1}$ also commute.

Problem 6

For which values of $a$ is $\begin{pmatrix} 1 & a & 0 \\ a & 1 & a \\ 0 & a & 1 \end{pmatrix}$ invertible?

12. Special Types of Invertible Matrices

Theorem 4.24: Special Matrix Inverses

For special matrix types:

Diagonal: $\text{diag}(d_1, \ldots, d_n)^{-1} = \text{diag}(1/d_1, \ldots, 1/d_n)$ if all $d_i \neq 0$
Orthogonal: $A^{-1} = A^T$ if $A^T A = I$
Triangular: Inverse of upper (lower) triangular is upper (lower) triangular
Involutory: $A^{-1} = A$ if $A^2 = I$

Example 4.9: Orthogonal Matrix Inverse

The rotation matrix $R_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ is orthogonal:

R_\theta^{-1} = R_\theta^T = \begin{pmatrix} \cos\theta & \sin\theta \\ -\sin\theta & \cos\theta \end{pmatrix} = R_{-\theta}

Rotating by $\theta$ then by $-\theta$ returns to the original position.

Example 4.10: Upper Triangular Inverse

For $U = \begin{pmatrix} 1 & 2 \\ 0 & 3 \end{pmatrix}$ :

U^{-1} = \begin{pmatrix} 1 & -2/3 \\ 0 & 1/3 \end{pmatrix}

The inverse is also upper triangular.

13. Connections to Other Topics

Determinants

$A$ is invertible iff $\det(A) \neq 0$ . Furthermore:

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

The adjugate formula: $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$

Eigenvalues

$A$ is invertible iff 0 is NOT an eigenvalue. For invertible $A$ :

\text{eigenvalues of } A^{-1} = \{1/\lambda : \lambda \text{ is eigenvalue of } A\}

Eigenvectors are the same for $A$ and $A^{-1}$ .

Linear Maps

$A$ is invertible iff the linear map it represents is an isomorphism:

Injective: $\ker(A) = \{0\}$
Surjective: $\text{im}(A) = F^n$

Change of Basis

If $P$ is the change of basis matrix, then:

[v]_{B'} = P^{-1}[v]_B

Change of basis matrices are always invertible.

14. Challenge Problems

Challenge 1: Sherman-Morrison Formula

Prove that if $A$ is invertible and $u, v$ are column vectors, then:

(A + uv^T)^{-1} = A^{-1} - \frac{A^{-1}uv^T A^{-1}}{1 + v^T A^{-1}u}

(provided $1 + v^T A^{-1}u \neq 0$ )

Challenge 2: Block Matrix Inverse

Find the inverse of $\begin{pmatrix} A & B \\ 0 & D \end{pmatrix}$ where $A$ and $D$ are invertible square matrices.

Challenge 3: Nilpotent Matrices

Prove that if $N$ is nilpotent ( $N^k = 0$ for some $k$ ), then $I - N$ is invertible with:

(I - N)^{-1} = I + N + N^2 + \cdots + N^{k-1}

15. Theoretical Insights

Theorem 4.25: General Linear Group

The set of all invertible $n \times n$ matrices over $F$ forms a group under multiplication, denoted $GL_n(F)$ :

Closure: Product of invertible matrices is invertible
Identity: $I_n$
Inverse: $A^{-1}$ exists for each $A \in GL_n(F)$
Associativity: Inherited from matrix multiplication

Remark 4.19: Notation

$GL_n(F)$ stands for "General Linear group". When $F = \mathbb{R}$ , we write $GL_n(\mathbb{R})$ ; when $F = \mathbb{C}$ , we write $GL_n(\mathbb{C})$ .

Theorem 4.26: Density of Invertible Matrices

Over $\mathbb{R}$ or $\mathbb{C}$ , the set of invertible matrices is "generic":

Singular matrices have measure zero
"Almost all" matrices are invertible
A random matrix with continuous entries is invertible with probability 1

Corollary 4.8: Perturbation

If $A$ is singular, then for almost any matrix $E$ , the perturbed matrix $A + \epsilon E$ is invertible for small $\epsilon \neq 0$ .

16. Study Tips

Check Dimensions

Only square matrices can be invertible. Before computing an inverse, verify the matrix is square.

Verify Your Answer

After computing $A^{-1}$ , always verify by checking $AA^{-1} = I$ .

Use 2×2 Formula

For $2 \times 2$ matrices, the formula $\frac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$ is fastest.

Remember Order Reversal

$(AB)^{-1} = B^{-1}A^{-1}$ . This is the same reversal that happens with transpose.

17. More Worked Examples

Example: Solving a System

Problem: Solve $\begin{pmatrix} 2 & 1 \\ 5 & 3 \end{pmatrix}\begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 3 \\ 8 \end{pmatrix}$ .

Solution: First find $A^{-1}$ :

A^{-1} = \frac{1}{6-5}\begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix} = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}

Then:

\begin{pmatrix} x \\ y \end{pmatrix} = A^{-1}b = \begin{pmatrix} 3 & -1 \\ -5 & 2 \end{pmatrix}\begin{pmatrix} 3 \\ 8 \end{pmatrix} = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

Example: Matrix Equation

Problem: Solve $AXB = C$ for $X$ where $A$ , $B$ , $C$ are known and $A$ , $B$ are invertible.

Solution:

AXB = C

A^{-1}(AXB) = A^{-1}C

XB = A^{-1}C

(XB)B^{-1} = A^{-1}CB^{-1}

X = A^{-1}CB^{-1}

Example: Power of Inverse

Problem: If $A^3 = I$ , find $A^{-1}$ .

Solution: From $A^3 = I$ :

A \cdot A^2 = I

This shows $A^2$ is the inverse of $A$ :

A^{-1} = A^2

Example: Inverse of Sum

Problem: If $AB = BA$ and both are invertible, show $(A + B)^{-1}$ commutes with $A$ .

Solution: We have $A(A + B) = A^2 + AB = A^2 + BA = (A + B)A$ .

If $A + B$ is invertible:

A(A + B)(A + B)^{-1} = (A + B)A(A + B)^{-1}

A = (A + B)A(A + B)^{-1}

Multiply by $(A + B)^{-1}$ on left:

(A + B)^{-1}A = A(A + B)^{-1}

18. Geometric Interpretation

Undoing Transformations

If $A$ represents a transformation (rotation, scaling, shear, etc.), then $A^{-1}$ represents the inverse transformation that "undoes" it:

If $A$ rotates by $\theta$ , then $A^{-1}$ rotates by $-\theta$
If $A$ scales by $k$ , then $A^{-1}$ scales by $1/k$
If $A$ shears, then $A^{-1}$ shears in the opposite direction

When Inversion Fails

A matrix is singular (not invertible) when it "collapses" dimension:

A projection onto a line in $\mathbb{R}^2$ is singular
A shear that maps everything to a single line is singular
Any transformation that loses information cannot be inverted

Determinant as Volume

The determinant $\det(A)$ represents the factor by which $A$ scales volumes:

$\det(A) = 0$ means $A$ collapses to lower dimension (not invertible)
$\det(A^{-1}) = 1/\det(A)$ restores the original volume

19. Historical Notes

Cayley's Contributions: Arthur Cayley developed the theory of matrix inverses in his 1858 paper. He recognized that invertibility corresponds to non-zero determinants and developed the adjugate formula.

Gaussian Elimination: While named after Carl Friedrich Gauss (early 1800s), the method was known to Chinese mathematicians nearly 2000 years earlier in texts like "The Nine Chapters on the Mathematical Art." It remains the standard algorithm for computing inverses.

General Linear Group: The group $GL_n(F)$ became central to modern mathematics through the work of Sophus Lie, Felix Klein, and others in the late 19th century. It connects linear algebra to group theory and geometry through the Erlangen program.

Numerical Computing: In the computer age, efficient and stable algorithms for matrix inversion became crucial. LU decomposition, developed by Alan Turing and others in the 1940s-50s, is now the standard approach for numerical work.

Modern Applications: Matrix inverses appear throughout science and engineering: solving systems of equations, computer graphics (inverting transformations), machine learning (solving linear regression), quantum mechanics, and control theory.

20. Left and Right Inverses

Definition 4.14: One-Sided Inverses

For a (possibly non-square) matrix $A$ :

Left inverse: $L$ such that $LA = I$
Right inverse: $R$ such that $AR = I$

Theorem 4.27: Existence of One-Sided Inverses

For an $m \times n$ matrix $A$ :

$A$ has a left inverse iff $A$ is injective (columns independent) iff $\text{rank}(A) = n$
$A$ has a right inverse iff $A$ is surjective (rows span $F^m$ ) iff $\text{rank}(A) = m$
$A$ has both iff $m = n$ and $A$ is invertible

Example 4.11: Left Inverse Example

Let $A = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ . A left inverse is any $L = \begin{pmatrix} 1 & c \end{pmatrix}$ for any $c$ :

LA = \begin{pmatrix} 1 & c \end{pmatrix}\begin{pmatrix} 1 \\ 0 \end{pmatrix} = 1

Left inverses are NOT unique for non-square matrices!

Remark 4.20: Moore-Penrose Pseudoinverse

The Moore-Penrose pseudoinverse $A^+$ generalizes the inverse to all matrices (including non-square and singular). It satisfies:

$AA^+A = A$
$A^+AA^+ = A^+$
$(AA^+)^T = AA^+$
$(A^+A)^T = A^+A$

Applications

Solving Linear Systems

$Ax = b \Rightarrow x = A^{-1}b$ . Theoretical but LU is faster in practice.

Computer Graphics

Inverse transformations for camera models, view projections, and animation.

Cryptography

Hill cipher uses matrix multiplication for encryption and inverse for decryption.

Statistics

Covariance matrix inverse in multivariate distributions and regression.

Key Takeaways

• Invertible ⟺ det ≠ 0 ⟺ full rank ⟺ columns independent
• 2×2 formula: swap diagonals, negate off-diagonals, divide by det
• Gauss-Jordan: row reduce [A|I] to [I|A⁻¹]
• (AB)⁻¹ = B⁻¹A⁻¹ — order reverses!

What's Next?

Now that you understand matrix inverses, the next topics build on this foundation:

Elementary Matrices: The building blocks that make Gaussian elimination work
Matrix Rank: The fundamental dimension that determines invertibility
Determinants: An algebraic criterion for invertibility

These concepts together form the theoretical backbone of computational linear algebra.

Matrix Inverse Practice

Questions

Correct

Accuracy

A^{-1}

exists, what is

AA^{-1}

Easy

Not attempted

Is the zero matrix invertible?

Easy

Not attempted

(AB)^{-1} = ?

Medium

Not attempted

A

n \times n

and

Ax = 0

has only

x = 0

as solution, is

A

invertible?

Medium

Not attempted

(A^{-1})^T = ?

Medium

Not attempted

3 \times 4

matrix is invertible:

Easy

Not attempted

\det(A) \neq 0

, then

A

is:

Medium

Not attempted

How do you compute

A^{-1}

using row reduction?

Medium

Not attempted

AB = I

for square matrices, is

BA = I

Hard

Not attempted

(A^{-1})^{-1} = ?

Easy

Not attempted

A

is invertible, is

A^T

invertible?

Medium

Not attempted

What is

\det(A^{-1})

Medium

Not attempted

Matrix Inverse

1. Definition and Motivation

2. One-Sided Inverses for Square Matrices

3. Characterizations of Invertibility

4. Properties of the Inverse

5. Computing the Inverse

6. Solving Systems with Inverses

7. Worked Examples

8. Common Mistakes

Mistake 1: Wrong Order for Product Inverse

Mistake 2: Assuming Non-Square Matrices Have Inverses

Mistake 3: Assuming Every Square Matrix is Invertible

Mistake 4: Inverse of Sum

9. Key Takeaways

Definition

Invertibility Test

Product Inverse

Computation

10. Quick Reference Summary

11. Additional Practice Problems

12. Special Types of Invertible Matrices

13. Connections to Other Topics

14. Challenge Problems

Challenge 1: Sherman-Morrison Formula

Challenge 2: Block Matrix Inverse

Challenge 3: Nilpotent Matrices

15. Theoretical Insights

16. Study Tips

Check Dimensions

Verify Your Answer

Use 2×2 Formula

Remember Order Reversal

17. More Worked Examples

18. Geometric Interpretation

Undoing Transformations

When Inversion Fails

Determinant as Volume

19. Historical Notes

20. Left and Right Inverses

Applications

Solving Linear Systems

Computer Graphics

Cryptography

Statistics

Related Topics

Key Takeaways

What's Next?