UNIT 8 • MATRICES & LINEAR ALGEBRA

Determinant
Cofactor Expansion & Row Reduction

The determinant $|A|$ of a square matrix is a single number: $ad - bc$ for a 2×2 matrix, and for bigger matrices you expand along a row using cofactors. If $|A| = 0$ , the matrix has no inverse.

HOW A DETERMINANT IS FOUNDSQUARE MATRICES ONLY

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \qquad |A| = \sum_{j=1}^{n} (-1)^{1+j}\, a_{1j}\, M_{1j}

1. Pick a RowUsually row 1 (or the row with the most zeros, so there is less to work out).

2. Find Each MinorCross out the entry's row and column;

M_{1j}

is the determinant of what is left.

3. Signs and SumUse the signs

+,\ -,\ +,\ \dots

and add

a_{1j} M_{1j}

with those signs.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

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

STEP 01 • EXPAND ALONG ROW 1

Each entry of row 1 multiplies the 2×2 determinant left after crossing out its row and column. The signs go + − +.

|A| = 2\begin{vmatrix} 0 & -1 \\ 4 & 5 \end{vmatrix} - (-3)\begin{vmatrix} 2 & -1 \\ 1 & 5 \end{vmatrix} + 1\begin{vmatrix} 2 & 0 \\ 1 & 4 \end{vmatrix}

STEP 02 • WORK OUT THE THREE MINORS

Each 2×2 determinant is $ad - bc$ :

M_{11} = (0)(5) - (-1)(4) = 4

M_{12} = (2)(5) - (-1)(1) = 11

M_{13} = (2)(4) - (0)(1) = 8

STEP 03 • COMBINE WITH THE SIGNS

|A| = 2(4) - (-3)(11) + 1(8) = 8 + 33 + 8 = 49

STEP 04 • CHECK BY ROW REDUCTION

Make the matrix upper triangular; the determinant is then the product of the diagonal.

R_2 \to R_2 - R_1,\quad R_3 \to R_3 - \tfrac{1}{2}R_1:\quad \begin{bmatrix} 2 & -3 & 1 \\ 0 & 3 & -2 \\ 0 & \tfrac{11}{2} & \tfrac{9}{2} \end{bmatrix}

R_3 \to R_3 - \tfrac{11}{6}R_2:\quad \begin{bmatrix} 2 & -3 & 1 \\ 0 & 3 & -2 \\ 0 & 0 & \tfrac{49}{6} \end{bmatrix}

|A| = (2)(3)\left(\tfrac{49}{6}\right) = 49 \ \checkmark

FINAL CONCLUSION

|A| = 49

|A| = 49 \neq 0

Because the determinant is not zero, this matrix has an inverse.

LABORATORY

Interactive Determinant Calculator

LABORATORY

Determinant Engine

Size of A:

A =

Method

Common Questions

What is the determinant of a matrix?

A single number worked out from a square matrix. For a 2×2 matrix it is

ad - bc

. It tells you whether the matrix has an inverse (only when

|A| \neq 0

) and how the matrix scales area or volume.

How do you find the determinant of a 3×3 matrix?

Expand along the first row: take each entry, multiply it by the 2×2 determinant left when you cross out its row and column (its minor), give the three terms the signs + − +, and add them. Row reduction to a triangular matrix gives the same answer.

What does a determinant of 0 mean?

The matrix is singular: it has no inverse, its rows are linearly dependent, and a system

Ax = b

with this matrix has either no solution or infinitely many.

What happens to the determinant when you swap two rows?

It changes sign. Adding a multiple of one row to another leaves it unchanged, and multiplying a row by k multiplies the determinant by k. These rules are what make row reduction work.

PreviousChi-Square Test

Unit 8•35 of 47

NextInverse of a Matrix

∑

f(x)∫dxπ

NETZ Engine

Evaluating numerical algorithm

UNIT 8 • MATRICES & LINEAR ALGEBRA

Determinant
Cofactor Expansion & Row Reduction

The determinant $|A|$ of a square matrix is a single number: $ad - bc$ for a 2×2 matrix, and for bigger matrices you expand along a row using cofactors. If $|A| = 0$ , the matrix has no inverse.

HOW A DETERMINANT IS FOUNDSQUARE MATRICES ONLY

\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \qquad |A| = \sum_{j=1}^{n} (-1)^{1+j}\, a_{1j}\, M_{1j}

1. Pick a RowUsually row 1 (or the row with the most zeros, so there is less to work out).

2. Find Each MinorCross out the entry's row and column;

M_{1j}

is the determinant of what is left.

3. Signs and SumUse the signs

+,\ -,\ +,\ \dots

and add

a_{1j} M_{1j}

with those signs.

Step-by-Step Worked Example

HAND-HOLDING TUTORIAL

OUR PROBLEM TO SOLVE

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

STEP 01 • EXPAND ALONG ROW 1

Each entry of row 1 multiplies the 2×2 determinant left after crossing out its row and column. The signs go + − +.

|A| = 2\begin{vmatrix} 0 & -1 \\ 4 & 5 \end{vmatrix} - (-3)\begin{vmatrix} 2 & -1 \\ 1 & 5 \end{vmatrix} + 1\begin{vmatrix} 2 & 0 \\ 1 & 4 \end{vmatrix}

STEP 02 • WORK OUT THE THREE MINORS

Each 2×2 determinant is $ad - bc$ :

M_{11} = (0)(5) - (-1)(4) = 4

M_{12} = (2)(5) - (-1)(1) = 11

M_{13} = (2)(4) - (0)(1) = 8

STEP 03 • COMBINE WITH THE SIGNS

|A| = 2(4) - (-3)(11) + 1(8) = 8 + 33 + 8 = 49

STEP 04 • CHECK BY ROW REDUCTION

Make the matrix upper triangular; the determinant is then the product of the diagonal.

R_2 \to R_2 - R_1,\quad R_3 \to R_3 - \tfrac{1}{2}R_1:\quad \begin{bmatrix} 2 & -3 & 1 \\ 0 & 3 & -2 \\ 0 & \tfrac{11}{2} & \tfrac{9}{2} \end{bmatrix}

R_3 \to R_3 - \tfrac{11}{6}R_2:\quad \begin{bmatrix} 2 & -3 & 1 \\ 0 & 3 & -2 \\ 0 & 0 & \tfrac{49}{6} \end{bmatrix}

|A| = (2)(3)\left(\tfrac{49}{6}\right) = 49 \ \checkmark

FINAL CONCLUSION

|A| = 49

|A| = 49 \neq 0

Because the determinant is not zero, this matrix has an inverse.

LABORATORY

Interactive Determinant Calculator

LABORATORY

Determinant Engine

Size of A:

A =

Method

Common Questions

What is the determinant of a matrix?

A single number worked out from a square matrix. For a 2×2 matrix it is

ad - bc

. It tells you whether the matrix has an inverse (only when

|A| \neq 0

) and how the matrix scales area or volume.

How do you find the determinant of a 3×3 matrix?

What does a determinant of 0 mean?

The matrix is singular: it has no inverse, its rows are linearly dependent, and a system

Ax = b

with this matrix has either no solution or infinitely many.

What happens to the determinant when you swap two rows?

It changes sign. Adding a multiple of one row to another leaves it unchanged, and multiplying a row by k multiplies the determinant by k. These rules are what make row reduction work.

PreviousChi-Square Test

Unit 8•35 of 47

NextInverse of a Matrix

Determinant Cofactor Expansion & Row Reduction

Step-by-Step Worked Example

|A| = 49

Interactive Determinant Calculator

Determinant Engine

Common Questions

NETZ Engine

Determinant Cofactor Expansion & Row Reduction

Step-by-Step Worked Example

|A| = 49

Interactive Determinant Calculator

Determinant Engine

Common Questions

Determinant
Cofactor Expansion & Row Reduction

Determinant
Cofactor Expansion & Row Reduction