UNIT 8 • MATRICES & LINEAR ALGEBRA

Inverse of a Matrix
Adjoint & Gauss-Jordan Methods

The inverse $A^{-1}$ of a square matrix is the matrix with $A A^{-1} = I$ . Find it as $A^{-1} = \frac{1}{|A|}\operatorname{adj} A.$ It exists only when $|A| \neq 0$ .

HOW THE INVERSE IS FOUNDNEEDS |A| ≠ 0

A^{-1} = \frac{1}{|A|}\,\operatorname{adj} A, \qquad \operatorname{adj} A = \left[ C_{ij} \right]^{T}, \quad C_{ij} = (-1)^{i+j} M_{ij}

1. DeterminantFind

|A|

. If it is 0, stop: there is no inverse.

2. CofactorsFor each entry,

C_{ij}

= its minor with sign

(-1)^{i+j}

3. Adjoint ÷ |A|Transpose the cofactor matrix to get

\operatorname{adj} A

, then divide every entry by

|A|

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $A^{-1}$ for $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$ .

STEP 01 • DETERMINANT

Expand along row 1:

|A| = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1

|A| = 1 ≠ 0, so the inverse exists.

STEP 02 • COFACTORS

C_{11} = -24,\quad C_{12} = 20,\quad C_{13} = -5

C_{21} = 18,\quad C_{22} = -15,\quad C_{23} = 4

C_{31} = 5,\quad C_{32} = -4,\quad C_{33} = 1

For example, $C_{12} = -\begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} = -(0 - 20) = 20$ .

STEP 03 • ADJOINT (TRANSPOSE THE COFACTORS)

\operatorname{adj} A = \begin{bmatrix} -24 & 20 & -5 \\ 18 & -15 & 4 \\ 5 & -4 & 1 \end{bmatrix}^{T} = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

STEP 04 • DIVIDE BY |A| AND CHECK

A^{-1} = \frac{1}{1}\operatorname{adj} A = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

\text{Row 1 of } A \cdot \text{column 1 of } A^{-1} = (1)(-24) + (2)(20) + (3)(-5) = 1 \ \checkmark

FINAL CONCLUSION

A⁻¹ = adj A, since |A| = 1

A^{-1} = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

Multiplying A by this matrix gives the identity matrix I.

LABORATORY

Interactive Inverse Calculator

LABORATORY

Matrix Inverse Engine

Size of A:

A =

Method

Common Questions

How do you find the inverse of a matrix?

Check

|A| \neq 0

, find the cofactor of every entry, transpose that cofactor matrix to get the adjoint, then divide by the determinant:

A^{-1} = \frac{1}{|A|}\operatorname{adj} A

. For bigger matrices, row-reduce

[A \mid I]

[I \mid A^{-1}]

(Gauss-Jordan).

What is the formula for the inverse of a 2×2 matrix?

Swap the diagonal entries, change the signs of the other two, and divide by the determinant:

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

When does a matrix have no inverse?

When its determinant is 0 (a singular matrix). Only square matrices can have an inverse.

What is the difference between the adjoint and the inverse?

The adjoint (adjugate) is the transpose of the cofactor matrix. The inverse is the adjoint divided by the determinant, so

A \cdot \operatorname{adj} A = |A|\, I

while

A \cdot A^{-1} = I

PreviousDeterminant

Unit 8•36 of 47

NextRank & Echelon Form

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 8 • MATRICES & LINEAR ALGEBRA

Inverse of a Matrix
Adjoint & Gauss-Jordan Methods

The inverse $A^{-1}$ of a square matrix is the matrix with $A A^{-1} = I$ . Find it as $A^{-1} = \frac{1}{|A|}\operatorname{adj} A.$ It exists only when $|A| \neq 0$ .

HOW THE INVERSE IS FOUNDNEEDS |A| ≠ 0

A^{-1} = \frac{1}{|A|}\,\operatorname{adj} A, \qquad \operatorname{adj} A = \left[ C_{ij} \right]^{T}, \quad C_{ij} = (-1)^{i+j} M_{ij}

1. DeterminantFind

|A|

. If it is 0, stop: there is no inverse.

2. CofactorsFor each entry,

C_{ij}

= its minor with sign

(-1)^{i+j}

3. Adjoint ÷ |A|Transpose the cofactor matrix to get

\operatorname{adj} A

, then divide every entry by

|A|

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

Find $A^{-1}$ for $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{bmatrix}$ .

STEP 01 • DETERMINANT

Expand along row 1:

|A| = 1(0 - 24) - 2(0 - 20) + 3(0 - 5) = -24 + 40 - 15 = 1

|A| = 1 ≠ 0, so the inverse exists.

STEP 02 • COFACTORS

C_{11} = -24,\quad C_{12} = 20,\quad C_{13} = -5

C_{21} = 18,\quad C_{22} = -15,\quad C_{23} = 4

C_{31} = 5,\quad C_{32} = -4,\quad C_{33} = 1

For example, $C_{12} = -\begin{vmatrix} 0 & 4 \\ 5 & 0 \end{vmatrix} = -(0 - 20) = 20$ .

STEP 03 • ADJOINT (TRANSPOSE THE COFACTORS)

\operatorname{adj} A = \begin{bmatrix} -24 & 20 & -5 \\ 18 & -15 & 4 \\ 5 & -4 & 1 \end{bmatrix}^{T} = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

STEP 04 • DIVIDE BY |A| AND CHECK

A^{-1} = \frac{1}{1}\operatorname{adj} A = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

\text{Row 1 of } A \cdot \text{column 1 of } A^{-1} = (1)(-24) + (2)(20) + (3)(-5) = 1 \ \checkmark

FINAL CONCLUSION

A⁻¹ = adj A, since |A| = 1

A^{-1} = \begin{bmatrix} -24 & 18 & 5 \\ 20 & -15 & -4 \\ -5 & 4 & 1 \end{bmatrix}

Multiplying A by this matrix gives the identity matrix I.

LABORATORY

Interactive Inverse Calculator

LABORATORY

Matrix Inverse Engine

Size of A:

A =

Method

Common Questions

How do you find the inverse of a matrix?

Check

|A| \neq 0

, find the cofactor of every entry, transpose that cofactor matrix to get the adjoint, then divide by the determinant:

A^{-1} = \frac{1}{|A|}\operatorname{adj} A

. For bigger matrices, row-reduce

[A \mid I]

[I \mid A^{-1}]

(Gauss-Jordan).

What is the formula for the inverse of a 2×2 matrix?

Swap the diagonal entries, change the signs of the other two, and divide by the determinant:

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

When does a matrix have no inverse?

When its determinant is 0 (a singular matrix). Only square matrices can have an inverse.

What is the difference between the adjoint and the inverse?

The adjoint (adjugate) is the transpose of the cofactor matrix. The inverse is the adjoint divided by the determinant, so

A \cdot \operatorname{adj} A = |A|\, I

while

A \cdot A^{-1} = I

PreviousDeterminant

Unit 8•36 of 47

NextRank & Echelon Form

Inverse of a Matrix Adjoint & Gauss-Jordan Methods

Step-by-Step Worked Example

A⁻¹ = adj A, since |A| = 1

Interactive Inverse Calculator

Matrix Inverse Engine

Common Questions

NETZ Engine

Inverse of a Matrix Adjoint & Gauss-Jordan Methods

Step-by-Step Worked Example

A⁻¹ = adj A, since |A| = 1

Interactive Inverse Calculator

Matrix Inverse Engine

Common Questions

Inverse of a Matrix
Adjoint & Gauss-Jordan Methods

Inverse of a Matrix
Adjoint & Gauss-Jordan Methods