If A = 2 3 4 5 , prove that A − A T is a skew-symmetric matrix.

Given: A = 2 3 4 5 A T = 2 4 3 5 Now, A - A T = 2 3 4 5 - 2 4 3 5 ⇒ A - A T = 2 - 2 3 - 4 4 - 3 5 - 5 ⇒ A - A T = 0 - 1 1 0 . . . . i A - A T T = 0 - 1 1 0 T ⇒ A - A T T = 0 1 - 1 0 ⇒ A - A T T = - 0 - 1 1 0 From equation i ⇒ A - A T T = - A - A T Thus, A - A T is a skew symmetric matrix.

If A = 3 - 4 1 - 1 , show that A − A T is a skew symmetric matrix.

Given: A = 3 - 4 1 - 1 A T = 3 1 - 4 - 1 Now, A - A T = 3 - 4 1 - 1 - 3 1 - 4 - 1 ⇒ A - A T = 3 - 3 - 4 - 1 1 + 4 - 1 + 1 ⇒ A - A T = 0 - 5 5 0 . . . i A - A T T = 0 - 5 5 0 T ⇒ A - A T T = 0 5 - 5 0 ⇒ A - A T T = - 0 - 5 5 0 From i equation, ⇒ A - A T T = - A - A T Thus, ( A - A T ) is a skew symmetric matrix.

If the matrix A = 5 2 x y z - 3 4 t - 7 is a symmetric matrix, find x , y , z and t .

Given: A = 5 2 x y z - 3 4 t - 7 ⇒ A T = 5 y 4 2 z t x - 3 - 7 Since A is a symmetric matrix, ( A T ) = A ⇒ 5 y 4 2 z t x - 3 - 7 = 5 2 x y z - 3 4 t - 7 The corresponding elements of two equal matrices are equal. ∴ x = 4 , y = 2 , z = z , t = - 3 Hence, x = 4 , y = 2 , t = - 3 and z can have any value.

Express the matrix A = 4 2 - 1 3 5 7 1 - 2 1 as the sum of a symmetric and a skew-symmetric matrix.

Given: A = 4 2 - 1 3 5 7 1 - 2 1 A T = 4 3 1 2 5 - 2 - 1 7 1 Every square matrix A can uniquely be expressed as the sum of a symmetric and skew-symmetric matrix i.e., A = 1 2 ( A + A T ) + 1 2 ( A − A T ) Let X = 1 2 A + A T = 1 2 4 2 - 1 3 5 7 1 - 2 1 + 4 3 1 2 5 - 2 - 1 7 1 ⇒ X = 1 2 8 5 0 5 10 5 0 5 2 ⇒ X = 4 5 2 0 5 2 5 5 2 0 5 2 1 X T = 4 5 2 0 5 2 5 5 2 0 5 2 1 T = 4 5 2 0 5 2 5 5 2 0 5 2 1 = X Let Y = 1 2 A - A T = 1 2 4 2 - 1 3 5 7 1 - 2 1 - 4 3 1 2 5 - 2 - 1 7 1 = 0 - 1 2 - 1 1 2 0 9 2 1 - 9 2 0 Y T = 0 - 1 2 - 1 1 2 0 9 2 1 - 9 2 0 T = 0 1 2 1 - 1 2 0 - 9 2 - 1 9 2 0 = - 0 - 1 2 - 1 1 2 0 9 2 1 - 9 2 0 = - Y Thus, X is a symmetric matrix and Y is a skew - symmetric matrix. Now, X + Y = 4 5 2 0 5 2 5 5 2 0 5 2 1 + 0 - 1 2 - 1 1 2 0 9 2 1 - 9 2 0 = 4 2 - 1 3 5 7 1 - 2 1 = A

Define a symmetric matrix. Prove that for A = 2 4 5 6 , A + A T is a symmetric matrix where A T is the transpose of A .

A square matrix A is called a symmetric matrix, if A T = A Given: A = 2 4 5 6 A T = 2 5 4 6 Now, A + A T = 2 4 5 6 + 2 5 4 6 ⇒ A + A T = 4 9 9 12 . . . 1 Now, A + A T T = 4 9 9 12 T = 4 9 9 12 = A + A T ∴ A + A T T = A + A T Thus, ( A + A T ) is a symmetric matrix.

Express the matrix A = 3 - 4 1 - 1 as the sum of a symmetric and a skew-symmetric matrix.

Given: A = 3 - 4 1 - 1 A T = 3 1 - 4 - 1 Every square matrix A can uniquely be expressed as sum of a symmetric and skew - symmetric matrix i.e., A = 1 2 ( A + A T ) + 1 2 ( A − A T ) . Let X = 1 2 A + A T = 1 2 3 - 4 1 - 1 + 3 1 - 4 - 1 = 3 - 3 2 - 3 2 - 1 X T = 3 - 3 2 - 3 2 - 1 T = 3 - 3 2 - 3 2 - 1 = X Let Y = 1 2 A - A T = 1 2 3 - 4 1 - 1 - 3 1 - 4 - 1 = 0 - 5 2 5 2 0 Y T = 0 - 5 2 5 2 0 T = 0 5 2 - 5 2 0 = - 0 - 5 2 5 2 0 = Y Therefore X is a symmetric matrix and Y is a skew - symmetric matrix. X + Y = 3 - 3 2 - 3 2 - 1 + 0 - 5 2 5 2 0 = 3 - 4 1 - 1 = A

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE

R D Sharma Solutions for Chapter: Algebra of Matrices, Exercise 1: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Algebra of Matrices, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 5: Algebra of Matrices, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Algebra of Matrices, Exercise 1: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 1

If $A = [\begin{matrix} 2 & 3 \\ 4 & 5 \end{matrix}]$ , prove that $A - A^{T}$ is a skew-symmetric matrix.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 2

If $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ , show that $A - A^{T}$ is a skew symmetric matrix.

EASY

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 3

If the matrix $A = [\begin{matrix} 5 & 2 & x \\ y & z & - 3 \\ 4 & t & - 7 \end{matrix}]$ is a symmetric matrix, find $x, y, z$ and $t$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 4

Let $A = [\begin{matrix} 3 & 2 & 7 \\ 1 & 4 & 3 \\ - 2 & 5 & 8 \end{matrix}]$ . Find matrices $X$ and $Y$ such that $X + Y = A$ , where $X$ is a symmetric and $Y$ is a skew-symmetric matrix.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 5

Express the matrix $A = [\begin{matrix} 4 & 2 & - 1 \\ 3 & 5 & 7 \\ 1 & - 2 & 1 \end{matrix}]$ as the sum of a symmetric and a skew-symmetric matrix.

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 6

Define a symmetric matrix. Prove that for $A = [\begin{matrix} 2 & 4 \\ 5 & 6 \end{matrix}]$ , $A + A^{T}$ is a symmetric matrix where $A^{T}$ is the transpose of $A$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 7

Express the matrix $A = [\begin{matrix} 3 & - 4 \\ 1 & - 1 \end{matrix}]$ as the sum of a symmetric and a skew-symmetric matrix.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 5 - Algebra of Matrices>EXERCISE>Q 8

Express the following matrix as the sum of a symmetric and skew-symmetric matrix and verify your result: $[\begin{matrix} 3 & - 2 & - 4 \\ 3 & - 2 & - 5 \\ - 1 & 1 & 2 \end{matrix}]$ .

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