Algebra

Matrices & Determinants

Medium

common

If $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ then $|adj(3A^T)|^2$ is equal to

$3^6$

$3^9$

$3^{12}$

$3^{18}$

✅ Correct Option: 3

We need to find $|adj(3A^T)|^2$ . Instead of calculating the adjoint directly, we'll use determinant properties which is much faster.

Key Properties:

$|A^T| = |A|$
$|kA| = k^n|A|$ where $k$ is a scalar and $n$ is the order of matrix
$|adj(B)| = |B|^{n-1}$ where $n$ is the order of matrix

First, find $|A|$ using the given matrix $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}$

Expanding along Row 3 (it has two zeros, making calculation easier):

$|A| = 0 \cdot (\text{cofactor}) - 0 \cdot (\text{cofactor}) + 1 \cdot \begin{vmatrix} 1 & 2 \\ 2 & 3 \end{vmatrix}$

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

Now find $|3A^T|$ :

Since $|A^T| = |A| = -1$ and $A$ is a $3 \times 3$ matrix:

$|3A^T| = 3^3 \cdot |A^T| = 27 \cdot (-1) = -27$

Why $3^3$ ? When we multiply a $3 \times 3$ matrix by $3$ , the determinant gets multiplied by $3$ three times.

Next, find $|adj(3A^T)|$ :

Using the property for a $3 \times 3$ matrix:

$|adj(3A^T)| = |3A^T|^{3-1} = |3A^T|^2 = (-27)^2 = 729$

Finally, find $|adj(3A^T)|^2$ :

$|adj(3A^T)|^2 = (729)^2 = 3^{12}$

The answer is $3^{12}$

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If A=[121231001]A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}A=​120​230​111​​ then ∣adj(3AT)∣2|adj(3A^T)|^2∣adj(3AT)∣2 is equal to

Solution

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CUET Mathematics Past Year Questions (Topic-Wise)

If $A = \begin{bmatrix} 1 & 2 & 1 \\ 2 & 3 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ then $|adj(3A^T)|^2$ is equal to