The trace of a square matrix is defined to be the sum of its diagonal entries. If is a matrix such that the trace of is 3 and the trace of is , then the value of the determinant of is _____. [2020]

Question

The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $- 18$ , then the value of the determinant of $A$ is _____. [2020]

Smart Achievers · Accepted Answer

(5)

$∵$ $The trace of A is 3$

Let $A = [\begin{matrix} x & y \\ z & 3 - x \end{matrix}];$ Now, $A^{2} = [\begin{matrix} x & y \\ z & 3 - x \end{matrix}] [\begin{matrix} x & y \\ z & 3 - x \end{matrix}]$

$= [\begin{matrix} x^{2} + y z & x y + 3 y - x y \\ x z + 3 z - x z & y z + 9 + x^{2} - 6 x \end{matrix}]$ $= [\begin{matrix} x^{2} + y z & 3 y \\ 3 z & y z + 9 + x^{2} - 6 x \end{matrix}]$

$\Rightarrow A^{3} = [\begin{matrix} x^{2} + y z & 3 y \\ 3 z & y z + 9 + x^{2} - 6 x \end{matrix}] [\begin{matrix} x & y \\ z & 3 - x \end{matrix}]$

$= [\begin{matrix} x^{3} + x y z + 3 y z & x^{2} y + y^{2} z + 9 y - 3 x y \\ 3 z x + y z^{2} + 9 z + x^{2} z - 6 x z & 6 y z + 27 + 3 x^{2} - 18 x - x y z - 9 x - x^{3} + 6 x^{2} \end{matrix}]$

Given that the trace of $A^{3}$ is $- 18$

$∴$ $x^{3} + x y z + 3 y z + 6 y z + 27 + 3 x^{2} - 18 x - x y z - 9 x - x^{3} + 6 x^{2} = - 18$

$\Rightarrow 9 y z + 9 x^{2} - 27 x + 27 = - 18$ $\Rightarrow y z + x^{2} - 3 x + 3 = - 2$

$\Rightarrow 3 x - x^{2} - y z = 5$ $\dots (i)$

Now, $| A |$ $= 3 x - x^{2} - y z$

Hence, $| A |$ $= 5$ $from (i)$

Solution

Q.
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $- 18$ , then the value of the determinant of $A$ is _____. [2020]

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Solution

Q. The trace of a square matrix is defined to be the sum of its diagonal entries. If A is a 2×2 matrix such that the trace of A is 3 and the trace of A3 is -18, then the value of the determinant of A is _____. [2020]

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Q.
The trace of a square matrix is defined to be the sum of its diagonal entries. If $A$ is a $2 \times 2$ matrix such that the trace of $A$ is 3 and the trace of $A^{3}$ is $- 18$ , then the value of the determinant of $A$ is _____. [2020]