Fill In The Blanks[1 Marks ]

Question 11 Mark

Fill in the blank.
If A and B are symmetric matrices of same order, then AB is symmetric if and only if _________.

Answer

If A and B are symmetric matrices of same order, then AB is symmetric if and only if AB = BA.
Solution:
If A and B are symmetric matrices of same order, then AB is symmetric if and only if AB = BA.
$\therefore$ (AB)'
= B'A' = BA $[\because$ AB = BA$]$
= AB

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Question 21 Mark

Fill in the blank.
Sum of two skew symmetric matrices is always _________ matrix.

Answer

Sum of two skew symmetric matrices is always skew-symmetric matrix.Solution:
Let A is a given matrix, then (-A) is skew-symmetric matrix. Similarly, for a given matrix (-B) is a skew-symmetric matrix. Hence, -A - B = -(A + B) ⇒ Sum of two skew-symmetric matrices is always skew-symmetric matrix.

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Question 31 Mark

Fill in the blank.
If A is skew symmetric, then kA is a _________. (k is any scalar)

Answer

If A is skew symmetric, then kA is a skew-symmetric. (k is any scalar)Solution:
Given A is skew-symmetric matrix $\therefore$ A' = -A $\therefore$ (kA)' = kA' = k(-A) = -kA $\therefore$ (kA) is also skew-symmetric matrix.

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Question 41 Mark

Fill in the blank.
Matrix multiplication is _________ over addition.

Answer

Matrix multiplication is distributive over addition.Solution:
Matrix multiplication is distributive over addition. Consider, three matrices A, B and C then

A(B + C) = AB + AC
(A + B)C = AC + BC

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Question 51 Mark

Fill in the blank.
If $A$ is a skew symmetric matrix, then $A^2$ is a _________.

Answer

If $A$ is a skew-symmetric matrix, then $A^2$ is a symmetric matrix.
$\because A' = -A$
$\therefore (A^2)' = A'^2$
$= (-A)^2 [\because A' = -A]$
$= A^2$
So, $A^2$ is a symmetric matrix.

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Question 61 Mark

Fill in the blank.
If $A$ is a symmetric matrix, then $A^3$ is a _________ matrix.

Answer

Given A is symmetric matrix
$\therefore A' = A$
Now $(A^3)' = (A')^3 [\because (A')^n = (A^n)'] = A^3$

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Question 71 Mark

Fill in the blank.
The negative of a matrix is obtained by multiplying it by _________.

Answer

The negative of a matrix is obtained by multiplying it by -1.Solution:
Let A be a matrix of any order. We can find the negative of A by multiplying all the elements of A by -1 i.e., -A = -1[A] Thus, the negative of a matrix is obtained by multiplying it by -1.

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Question 81 Mark

Fill in the blank.
In applying one or more row operations while finding $A^{-1}$ by elementary row operations, we obtain all zeros in one or more, then $A^{-1}$ _________.

Answer

In applying one or more row operations while finding $A^{-1}$ by elementary row operations, we obtain all zeros in one or more, then $A^{-1}$ does not exist.

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Question 91 Mark

Fill in the blank.
The product of any matrix by the scalar _________ is the null matrix.

Answer

The product of any matrix by the scalar 0 is the null matrix.Solution:
The product of any matrix by the scalar 0 is the null matrix. i.e., 0. A = 0 [where, A is any matrix]

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Question 101 Mark

Fill in the blank.
_________ matrix is both symmetric and skew symmetric matrix.

Answer

Null matrix is both symmetric and skew symmetric matrix.

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Question 111 Mark

Fill in the blank.
If A and B are square matrices of the same order, then:

(AB)' = _________.
(kA)' = _________. (k is any scalar)
[k(A - B)]' = _________.

Answer

If A and B are square matrices of the same order, we have:

(AB)' = B'A
(kA)' = kA' (k is any scalar)
[k(A - B)]' = k(A' - B')

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Question 121 Mark

Fill in the blank.
If A and B are symmetric matrices, then:

AB – BA is a _________.
BA – 2AB is a _________.

Answer

If A and B are symmetric matrices, then:

AB – BA is a skew-symmetric matrix.
BA – 2AB is a neither symmetric nor skew-symmetric matrix.

Solution:

AB - BA is a skew-symmetric matrix.

Since, [AB - BA]' = (AB') - (BA)'
= B'A' - A'B' $[\because$ (AB)' = B'A'$]$
= BA - AB $[\because$ A' = A and B' = B$]$
= -[AB - BA]
So, [AB - BA] is a skew-symmetric matrix.

[BA - 2AB] is a neither symmetric nor skew-symmetric matrix.

$\therefore$ (BA - 2AB)' = (BA)' - 2(AB)'
= A'B' - 2B'A'
= AB - 2BA
= -(2BA - AB)
So, [BA - 2AB] is neither symmetric nor skew-symmetric matrix.

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Question 131 Mark

Fill in the blank.
If A is symmetric matrix, then B′AB is _________.

Answer

If A is symmetric matrix, then B′AB is symmetric matrix.Solution:
Given A is symmetric matrix
$\therefore$ A' = A Now [B'AB]' = [B'(AB)]' = (AB)'(B')' $[\because$ (AB)' = B'A'$]$ = B'A'B = [B'AB] $[\because$ A' = A$]$ So, B'AB is a symmetric matrix.

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Question 141 Mark

Fill in the blank.
A matrix which is not a square matrix is called a _________ matrix.

Answer

A matrix which is not a square matrix is called a rectangular matrix.
For example a rectangular matrix is $A = [a_{ij}]_{m\times n},$
where $\text{m}\neq\text{n}.$

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MATHS Fill In The Blanks[1 Marks ]

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