SEE 2080 SET A ( General )

Subscribe My YouTube Channel Ambik Sir

13. If the matrices \( A = \begin{bmatrix} 2 & 1 \\ 3 & x \end{bmatrix} \) , \( B = \begin{bmatrix} 7 & 4 \\ 3 & 2 \end{bmatrix} \) and \( |AB|=10 \) , find the value of \( x \).

Solution:

\begin{align*} AB & = \begin{bmatrix} 2 & 1 \\ 3 & x \end{bmatrix} \begin{bmatrix} 7 & 4 \\ 3 & 2 \end{bmatrix} \\ & = \begin{bmatrix} 14+ 3  & 8 + 2 \\ 21 + 3x & 12 + 2x  \end{bmatrix} \\ & = \begin{bmatrix} 17  & 10 \\ 21 + 3x & 12 + 2x  \end{bmatrix} \\ \text{Now, } |AB|  & = \begin{vmatrix} 17  & 10 \\ 21 + 3x & 12 + 2x  \end{vmatrix} \\ \text{or, } 10 & = 17 (12 + 2x) - 10 ( 21 + 3x) \\ \text{or, } 10 & = 204 + 34 x - 210 - 30x = 10 \\ \text{or, } 4x  - 6 & = 10 \\ \text{or, } 4x & = 10 + 6 \\ \text{or, } 4x & = 16 \\ \text{or, } x & = \frac{16}{4} \\ \therefore x & = 4 \end{align*}