Question

Find the value of the deteaminats
3 -4 5
1 1 2
2 3 1
​

Answer 1

-22 is the determinant of the given matrix.

Step-by-step explanation:

Given: Matrix $A = \left[\begin{array}{ccc}3&-4&5\\1&1&2\\2&3&1\end{array}\right]$

To Find: Determinant |A| of the given matrix A

Solution:

• Determinant of the matrix

Determinant of any matrix can be calculated as,

$\begin{vmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{vmatrix} = a_{11} \begin{vmatrix}a_{22}&a_{23}\\a_{32}&a_{33}\end{vmatrix} - a_{12} \begin{vmatrix}a_{21}&a_{23}\\a_{31}&a_{33}\end{vmatrix} + a_{13} \begin{vmatrix}a_{21}&a_{22}\\a_{31}&a_{32}\end{vmatrix}$

$= a_{11} (a_{22}a_{33} - a_{23}a_{32}) - a_{12} (a_{21}a_{33} - a_{23}a_{31}) + a_{13} (a_{21}a_{32} - a_{22}a_{31})$

• Finding the determinant of the matrix A

We have given the matrix,

$A = \left[\begin{array}{ccc}3&-4&5\\1&1&2\\2&3&1\end{array}\right]$

Such that the determinant is

$|A| = 3 ((1 \times 1 ) - (2 \times 3)) - (-4) ((1 \times 1 ) - (2 \times 2)) + 5 ((1 \times 3 ) - (1 \times 2))$

$\Rightarrow |A| = 3 (-5) + 4 (-3) + 5 (1) = -22$

Hence, the determinant |A| of the matrix A is -22