SATHEE: Matrices and Determinants 1 Question 1

Matrices and Determinants 1 Question 1

1. If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that

$A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]$ , then $A B$ is equal to

(a) $[\begin{matrix} - 4 & - 2 \\ - 1 & 4 \end{matrix}]$

(b) $[\begin{matrix} 4 & - 2 \\ - 1 & - 4 \end{matrix}]$

(d) $[\begin{matrix} - 4 & 2 \\ 1 & 4 \end{matrix}]$

(2019 Main, 12 April I)

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Answer:

Correct Answer: 1. (b)

Solution:

Given matrix $A$ is a symmetric and matrix $B$ is a skew-symmetric.

$∴ A^{T} = A$ and $B^{T} = - B$

Since, $A + B = [\begin{matrix} 2 & 3 \\ 5 & - 1 \end{matrix}]$ (given)… (i)

On taking transpose both sides, we get

$\begin{aligned} (A + B)^{T} & = [\begin{array}{c} 2 & 3 \\ 5 & - 1 \end{array}] \\ \Rightarrow A^{T} + B^{T} & = [\begin{array}{c} 2 & 5 \\ 3 & - 1 \end{array}] \end{aligned}$

Given, $A^{T} = A$ and $B^{T} = - B$

$\Rightarrow A - B = [\begin{matrix} 2 & 5 \\ 3 & - 1 \end{matrix}]$

On solving Eqs. (i) and (ii), we get

$\begin{aligned} A & = [\begin{array}{c} 2 & 4 \\ 4 & - 1 \end{array}] and B = [\begin{array}{c} 0 & - 1 \\ 1 & 0 \end{array}] \\ So, A B & = [\begin{array}{c} 2 & 4 & 0 & - 1 \\ 4 & - 1 & 1 & 0 \end{array}] = [\begin{array}{c} 4 & - 2 \\ - 1 & - 4 \end{array}] \end{aligned}$

Matrices and Determinants 1 Question 1

1. If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that

Answer:

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Matrices and Determinants 1 Question 1

1. If A is a symmetric matrix and B is a skew-symmetric matrix such that

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

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1. If $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix such that