Given the matrix: A = [[2, 1, 3], [0, −1, 2], [4, 5, 1]]. 2.2.1 Calculate the determinant of matrix A. 2.2.2…

Question

Given the matrix: A = [[2, 1, 3], [0, −1, 2], [4, 5, 1]]. 2.2.1 Calculate the determinant of matrix A. 2.2.2 Determine the cofactor of element a23.

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Here are the solutions for the matrix problems:

Given the matrix:

A = \begin{bmatrix} 2 & 1 & 3 \\ 0 & -1 & 2 \\ 4 & 5 & 1 \end{bmatrix}

2.2.1 Calculate the determinant of matrix A.

Use the cofactor expansion method.
We will expand along the first row.
The formula for the determinant of a $3 \times 3$ matrix is $\det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13}$ , where $C_{ij} = (-1)^{i+j}M_{ij}$ .
$\det(A) = 2 \cdot (-1)^{1+1} \det \begin{bmatrix} -1 & 2 \\ 5 & 1 \end{bmatrix} + 1 \cdot (-1)^{1+2} \det \begin{bmatrix} 0 & 2 \\ 4 & 1 \end{bmatrix} + 3 \cdot (-1)^{1+3} \det \begin{bmatrix} 0 & -1 \\ 4 & 5 \end{bmatrix}$
Calculate the $2 \times 2$ determinants (minors).
$\det \begin{bmatrix} -1 & 2 \\ 5 & 1 \end{bmatrix} = (-1)(1) - (2)(5) = -1 - 10 = -11$ $\det \begin{bmatrix} 0 & 2 \\ 4 & 1 \end{bmatrix} = (0)(1) - (2)(4) = 0 - 8 = -8$ $\det \begin{bmatrix} 0 & -1 \\ 4 & 5 \end{bmatrix} = (0)(5) - (-1)(4) = 0 - (-4) = 4$
Substitute the minor values back into the determinant formula and simplify.
$\det(A) = 2(1)(-11) + 1(-1)(-8) + 3(1)(4)$ $\det(A) = -22 + 8 + 12$ $\det(A) = -22 + 20$ $\det(A) = -2$
The determinant of matrix A is $\boxed{-2}$ .

2.2.2 Determine the cofactor of element $a_{23}$ .
Identify the element $a_{23}$ .
It is the element in the 2nd row and 3rd column, which is 2.
The formula for the cofactor $C_{ij}$ is $C_{ij} = (-1)^{i+j}M_{ij}$ , where $M_{ij}$ is the minor.
For $a_{23}$ , we need to find $C_{23}$ .
Determine the minor $M_{23}$ by deleting the 2nd row and 3rd column of matrix A.
$M_{23} = \det \begin{bmatrix} 2 & 1 \\ 4 & 5 \end{bmatrix}$
Calculate the determinant of the $2 \times 2$ minor matrix.
$M_{23} = (2)(5) - (1)(4) = 10 - 4 = 6$
Calculate the cofactor $C_{23}$ .
$C_{23} = (-1)^{2+3} M_{23}$ $C_{23} = (-1)^5 (6)$ $C_{23} = -1 \cdot 6$ $C_{23} = -6$
The cofactor of element $a_{23}$ is $\boxed{-6}$ .

Given the matrix: A = [[2, 1, 3], [0, -1, 2], [4, 5, 1]]. 2.2.1 Calculate the determinant of matrix A. 2.2.2 Determine the cofactor of element a23.

Answer

Step-by-step solution

Use the cofactor expansion method.

Calculate the $2 \times 2$ determinants (minors).

Substitute the minor values back into the determinant formula and simplify.

Identify the element $a_{23}$ .

Determine the minor $M_{23}$ by deleting the 2nd row and 3rd column of matrix A.

Calculate the determinant of the $2 \times 2$ minor matrix.

Calculate the cofactor $C_{23}$ .

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