SATHEE: Matrices and Determinants 2 Question 28

Matrices and Determinants 2 Question 28

31. Let $p λ^{4} + q λ^{3} + r λ^{2} + s λ + t$

$= | \begin{matrix} λ^{2} + 3 λ & λ - 1 & λ + 3 \\ λ + 1 & - 2 λ & λ - 4 \\ λ - 3 & λ + 4 & 3 λ \end{matrix} |$ be an identity in $λ$ , where $p, q, r, s$ and $t$ are constants. Then, the value of $t$ is…. .

(1981, 2M)

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Answer:

Correct Answer: 31. (0)

Solution:

Given, $| \begin{matrix} λ^{2} + 3 λ & λ - 1 & λ + 3 \\ λ + 1 & - 2 λ & λ - 4 \\ λ - 3 & λ + 4 & 3 λ \end{matrix} |$

$= p λ^{4} + q λ^{3} + r λ^{2} + s λ + t$

Thus, the value of $t$ is obtained by putting $λ = 0$ .

$\begin{aligned} \Rightarrow | \begin{array}{c} 0 & - 1 & 3 \\ 1 & 0 & - 4 \\ - 3 & 4 & 0 \end{array} | = t \\ \Rightarrow t = 0 \end{aligned}$

$[∵$ determinants of odd order skew-symmetric matrix is zero]

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Matrices and Determinants 2 Question 28

31. Let pλ4+qλ3+rλ2+sλ+t

Answer:

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31. Let $p λ^{4} + q λ^{3} + r λ^{2} + s λ + t$