GATE Civil 2024 Set 2 | Question: 38

Question

Consider two matrices $\mathbf{A}=\left[\begin{array}{lll}2 & 1 & 4 \\ 1 & 0 & 3\end{array}\right]$ and $\mathbf{B}=\left[\begin{array}{cc}-1 & 0 \\ 2 & 3 \\ 1 & 4\end{array}\right]$.
The determinant of the matrix  $\mathbf{AB}$ is $\_\_\_\_\_\_\_$ (in integer).

Answer 1

10

Answer 2

A is 2 x 3 and B is 3 x 2  matrix 

So,

AB is 2 x 2 

and its determinant exists.

Compute AB  =  [ 4 19 ]

                          [  2  12]  

then AB = 48 -38 =10

Answer 3

The determinant of a matrix is only defined for square matrices ($n \times n$). If the product of two matrices $AB$ results in a square matrix, its determinant can be calculated using standard expansion methods.

Given $A = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix}$ ($2 \times 3$) and $B = \begin{bmatrix} -1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix}$ ($3 \times 2$).

Step 1: Calculate the product } $AB$:
\[ AB = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 0 & 3 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 2 & 3 \\ 1 & 4 \end{bmatrix} \]
\[ AB = \begin{bmatrix} (2)(-1)+(1)(2)+(4)(1) & (2)(0)+(1)(3)+(4)(4) \\ (1)(-1)+(0)(2)+(3)(1) & (1)(0)+(0)(3)+(3)(4) \end{bmatrix} \]
\[ AB = \begin{bmatrix} -2+2+4 & 0+3+16 \\ -1+0+3 & 0+0+12 \end{bmatrix} = \begin{bmatrix} 4 & 19 \\ 2 & 12 \end{bmatrix} \]

Step 2: Calculate $\text{det}(AB)$:
\[ \text{det}(AB) = (4 \times 12) - (19 \times 2) \]
\[ \text{det}(AB) = 48 - 38 = 10 \]