Determinant of a matrices

Determinant of a matrix

A number that represents the magnitude of a square matrix is called a determinant. The determinant of matrix A is denoted by det(A) or |A|. It can be calculated provided that A is a square matrix.

If det(A) = 0, then matrix A is called a singular matrix, and if det(A) ≠ 0, then the matrix is called a non-singular matrix.

Determinant of a 2×2 matrix

Let A be a 2×2 matrix given by:

$$A = \begin{bmatrix} a & c \\ b & d \end{bmatrix}$$

Then, |A| = ad - bc. Therefore, det(A) = ad - bc.

That is, the determinant of a matrix is obtained by subtracting the product of the entries on the main diagonal from the product of entries in the leading diagonal.

Example 10.11: Find the determinant of matrix A, where:

$$A = \begin{bmatrix} 4 & 2 \\ 3 & 5 \end{bmatrix}$$

Solution: Given $A = \begin{bmatrix} 4 & 2 \\ 3 & 5 \end{bmatrix}$.

|A| = (4 × 5) - (3 × 2) = 20 - 6

∴ |A| = 14

Determinant of a 3×3 matrix

To find the determinant of a 3×3 matrix, the knowledge of minors and cofactors is necessary.

Minors

The minor of an element is the determinant of a sub-matrix formed by deleting the row and column of that element. Each element in a 3×3 matrix has its own minor. Minors are denoted by $M_{ij}$ where i and j are the row and column numbers, respectively.

Let matrix:

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

Examples of minors:

$$M_{11} = \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33} \end{vmatrix} = a_{22}a_{33} - a_{23}a_{32}$$ $$M_{12} = \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33} \end{vmatrix} = a_{21}a_{33} - a_{23}a_{31}$$

And so on for other minors.

Co-factors

A cofactor of an element $a_{ij}$ is defined by multiplying $(-1)^{i+j}$ by the minor of that element. The cofactor of matrix A is abbreviated as Cof(A).

If:

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{bmatrix}$$

Then, the cofactors are calculated as:

$$C_{ij} = (-1)^{i+j} \cdot M_{ij}$$

The pattern of signs for cofactors is:

$$\begin{bmatrix} + & - & + \\ - & + & - \\ + & - & + \end{bmatrix}$$

Example 10.13: Find the cofactors of element $b_{22}$ and $b_{32}$ in matrix B below:

$$B = \begin{bmatrix} 3 & 2 & 1 \\ 1 & 4 & 2 \\ 2 & 0 & 2 \end{bmatrix}$$

Solution:

Minor of $b_{22}$ (4): $M_{22} = \begin{vmatrix} 3 & 1 \\ 2 & 2 \end{vmatrix} = (3\cdot2)-(1\cdot2) = 4$

Cofactor of $b_{22}$: $(-1)^{2+2} \cdot 4 = 4$

Minor of $b_{32}$ (0): $M_{32} = \begin{vmatrix} 3 & 1 \\ 1 & 2 \end{vmatrix} = (3\cdot2)-(1\cdot1) = 5$

Cofactor of $b_{32}$: $(-1)^{3+2} \cdot 5 = -5$

Finding determinants of 3×3 matrices using cofactors

Choose any row or column. Calculate the cofactor of each element in the chosen row or column. The determinant is the sum of the products of each element and its cofactor.

Example 10.14: Find the determinant of matrix A:

$$A = \begin{bmatrix} 3 & 4 & 5 \\ 2 & -3 & 7 \\ 1 & 3 & 8 \end{bmatrix}$$

Solution (using row 1):

$|A| = 3 \begin{vmatrix} -3 & 7 \\ 3 & 8 \end{vmatrix} - 4 \begin{vmatrix} 2 & 7 \\ 1 & 8 \end{vmatrix} + 5 \begin{vmatrix} 2 & -3 \\ 1 & 3 \end{vmatrix}$

$|A| = 3(-24-21) - 4(16-7) + 5(6-(-3))$

$|A| = 3(-45) - 4(9) + 5(9) = -135 - 36 + 45 = -126$

Identity and inverse of a matrix

The inverse of a matrix

The inverse of a matrix A is denoted by $A^{-1}$. The inverse can only be defined for square matrices.

Properties of inverse of matrices

1. $AA^{-1} = A^{-1}A = I$, where I is the identity matrix.
2. $(A^{-1})^{-1} = A$
3. The inverse of matrix A exists if and only if A is a non-singular matrix, i.e., |A| ≠ 0.
4. When |A| = 0, then A is a singular matrix. Hence, $A^{-1}$ is undefined.
5. $(AB)^{-1} = B^{-1}A^{-1}$

Inverse of a 2×2 matrix

Let $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Then, det(A) = ad - cb. The inverse of A is given by:

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

Example 10.15: Find the inverse of the matrix:

$$A = \begin{bmatrix} 2 & 3 \\ 4 & 7 \end{bmatrix}$$

Solution: det(A) = (27) - (34) = 14 - 12 = 2

$$A^{-1} = \frac{1}{2} \begin{bmatrix} 7 & -3 \\ -4 & 2 \end{bmatrix}$$