Q72 / 180 · COMEDK 2026 (9 May Shift 2) · Mathematics

Given $A = \begin{bmatrix} x & 1 & -2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$. If $ABA^t = [-20]$ then the value of $x$ is:

  • A
    $1$
  • B
    $-1$
  • C
    $-3$
  • Correct answer
    D
    $11$

Solution

$A = \begin{bmatrix} x & 1 & -2 \end{bmatrix}$

$B = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

$AB = \begin{bmatrix} x & 1 & -2 \end{bmatrix} \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$

$AB = \begin{bmatrix} x(1) + 1(4) - 2(7) & x(2) + 1(5) - 2(8) & x(3) + 1(6) - 2(9) \end{bmatrix}$

$AB = \begin{bmatrix} x - 10 & 2x - 11 & 3x - 12 \end{bmatrix}$

$ABA^t = \begin{bmatrix} x - 10 & 2x - 11 & 3x - 12 \end{bmatrix} \begin{bmatrix} x \\ 1 \\ -2 \end{bmatrix}$

$ABA^t = [x(x - 10) + 1(2x - 11) - 2(3x - 12)]$

$ABA^t = [x^2 - 10x + 2x - 11 - 6x + 24]$

$ABA^t = [x^2 - 14x + 13]$

Given $ABA^t = [-20]$, we have:

$x^2 - 14x + 13 = -20$

$x^2 - 14x + 33 = 0$

$(x - 3)(x - 11) = 0$

$x = 3$ or $x = 11$

From the given options, $x = 11$.

Answer: $11$

Hence, the correct answer is option: 'D'.

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