## How to Find Determinants of a Matrix

### What is the Determinant of a Matrix?

The determinant of a matrix is a unique number that is only defined for square matrices (matrices with the same number of rows and columns). A determinant is used in calculus and other algebra-related matrices. It is a real number representing the matrix and can be used to solve a system of linear equations and find the inverse of a matrix.

### How to find the Determinant of a Matrix?

The following steps can be used to find the value of a matrix's determinant:

• Get the cofactor for each element in the first row or column.
• Multiply each element by the determinant of the corresponding cofactor.
• Add the results with the signs swapped.

As a starting point, the value of a matrix's determinant of a $$1 \times 1$$ matrix is the single value itself.
The cofactor of an element is a matrix that can be made by taking that element's row and column out of that Matrix.

### Determinant of 2 × 2 Matrix:

$$\left[ \begin{matrix} a & b\\ c & d\\ \end{matrix} \right]$$ $$|A|=a\ d\ -\ b\ c$$

### Determinant of 3 × 3 Matrix:

$$\left[ \begin{matrix} a & b &c \\ d & e & f \\ g & h & i \\ \end{matrix} \right]$$  $$|A|=ad-bc= a\ (e\ i \ –\ f\ h) \ –\ b\ (d\ i\ -\ f\ g) \ +\ c\ (d\ h \ –\ e\ g)$$

### Properties of Determinants of a Matrix

●    A determinant can only be used with a square matrix ($$1 \times 1$$, $$2 \times 2$$, $$3 \times 3$$, $$4 \times 4$$,...).
●    A determinant can be a real or complex number.
●    $$|A|$$ doesn't show the modulus of $$A$$ in this case; it shows the determinant of matrix $$A$$.
●    If the $$2 \times 2$$ matrix's elements are all the same, the determinant will be $$0$$.
●    If every number in a row or column of a $$2 \times 2$$ matrix is $$0$$, then the determinant is also $$0$$.
●    The determinant of the product of two matrices is the same as the product of their determinants. $$|AB| \ = \ |A| \ |B|$$.

### Exercises for Finding Determinants of a Matrix

1) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 4 & -8 \\\ -2 & 3 \end{bmatrix}$$

2) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 7 & 3 \\\ 1 & 6 \end{bmatrix}$$

3) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 5 & 4 \\\ 2 & -9 \end{bmatrix}$$

4) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 0 & -9 \\\ -4 & 1 \end{bmatrix}$$

5) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 5 & 7 \\\ 3 & 2 \end{bmatrix}$$

6) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 9 & 8 \\\ 11 & 7 \end{bmatrix}$$

7) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 12 & 6 \\\ -4 & 7 \end{bmatrix}$$

8) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 6 & 11 \\\ 9 & 13 \end{bmatrix}$$

9) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} -5 & 17 \\\ 3 & -4 \end{bmatrix}$$

10) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 4 & -1 & 8 \\\ 1 & 5 & -2 \\\ -3 & 4 & 6 \end{bmatrix}$$

1) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 4 & -8 \\\ -2 & 3 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 4 \times 3 \ - \ (-8) \times (-2) \ = \ 12 \ - \ 16 \ = \ -4}$$

2) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 7 & 3 \\\ 1 & 6 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 7 \times 6 \ - \ 3 \times 1 \ = \ 42 \ - \ 3 \ = \ 39}$$

3) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 5 & 4 \\\ 2 & -9 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -2 \times (-3) \ - \ 6 \times 5 \ = \ 6 \ - \ 30 \ = \ -24}$$

4) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 0 & -9 \\\ -4 & 1 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 0 \times 1 \ - \ (-9) \times (-4) \ = \ -36}$$

5) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 5 & 7 \\\ 3 & 2 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 5 \times 2 \ - \ 7 \times 3 \ = \ 10 \ - \ 21 \ = \ -11}$$

6) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 9 & 8 \\\ 11 & 7 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 9 \times 7 \ - \ 8 \times 11 \ = \ 63 \ - \ 88 \ = \ -25}$$

7) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 12 & 6 \\\ -4 & 7 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 12 \times 7 \ - \ 6 \times (-4) \ = \ 84 \ - \ (-24) \ = \ 108}$$

8) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 6 & 11 \\\ 9 & 13 \end{bmatrix}$$

$$\color{red}{|A| \ = \ 6 \times 13 \ - \ 11 \times 9 \ = \ 78 \ - \ 99 \ = \ -21}$$

9) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} -5 & 17 \\\ 3 & -4 \end{bmatrix}$$

$$\color{red}{|A| \ = \ -5 \times (-4) \ - \ 17 \times 3 \ = \ 20 \ - \ 51 \ = \ -31}$$

10) Find the determinant of the matrix: $$A \ = \ \begin{bmatrix} 4 & -1 & 8 \\\ 1 & 5 & -2 \\\ -3 & 4 & 6 \end{bmatrix}$$

$$\color{red}{det \begin{bmatrix} 5 & -2 \\\ 4 & 6 \end{bmatrix} \ = \ 38}$$
$$\color{red}{det \begin{bmatrix} 1 & -2 \\\ -3 & 6 \end{bmatrix} \ = \ 0}$$
$$\color{red}{det \begin{bmatrix} 1 & 5 \\\ -3 & 4 \end{bmatrix} \ = \ 19}$$
$$\color{red}{|A| \ = \ 4 \times 38 \ - \ (-1) \times 0 \ + \ 8 \times 19 \ = \ 304}$$

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