# Extra questions on Algebra of Matrices for Class 12

## True and False

Question 1
True and False
(a) A matrix denotes a number.
(b)Matrices of any order can be added.
(c)Two matrices are equal if they have same number of rows and same number of columns.
(d)Matrices of different order can not be subtracted.
(e)Matrix addition is associative as well as commutative.
(f)Matrix multiplication is commutative.
(g)A square matrix where every element is unity is called an identity matrix.
(h)If A and B are two square matrices of the same order, then A + B = B + A.
(i)If A and B are two matrices of the same order, then A – B = B – A.
(j)If matrix AB = O, then A = O or B = O or both A and B are null matrices.
(k)Transpose of a column matrix is a column matrix.
(l)If A and B are two square matrices of the same order, then AB = BA.
(m)If each of the three matrices of the same order are symmetric, then their sum is a symmetric matrix.
(n)If A and B are any two matrices of the same order, then (AB)2= A2 B2
(o)If (AB)2 = B2 A2, where A and B are not square matrices, then number of rows in A is equal to number of columns in B and number of columns in A is equal to number of rows in B.
(p)If A, B and C are square matrices of same order, then AB = AC always implies that B = C.
(q)AA, is always a symmetric matrix for any matrix A.
(r).If
$A=\begin{bmatrix} 2 & 3 & 1 \\ 1 & 4 & 2 \\ \end{bmatrix}$
and
$B=\begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \\ \end{bmatrix}$ , then AB and BA are defined and equal.
(s).If A is skew symmetric matrix, then A2 is a symmetric matrix.
(t).(AB)-1 = A-1. B-1 where A and B are invertible matrices satisfying commutative property with respect to multiplication.

(a)False
(b)False
(c)False
(d)True
(e)True
(f)False
(g) False
(h)True
(i)False
(j) False
(k)False
(l)False
(m)True
(n)False
(o)True
(p)False
(q)True
(r)False
(s)True
(t)True

Question 2
Which sums can be made from the following matrices ?

$A= \begin{bmatrix} 1 & 2 & 3 \\ \end{bmatrix}$
$B= \begin{bmatrix} 1 & 3 \\ -7 & 1 \\ \end{bmatrix}$
$C= \begin{bmatrix} 1 & -1 & 11 \\ 2 & -2 & 8 \\ 3 & -6 & 0 \\ \end{bmatrix}$
$D= \begin{bmatrix} 1 & 3 \\ -1 & 1 \\ \end{bmatrix}$
(a) A+A
(b) A+B
(c) A+C
(d) A+D
(e) B+A
(f) B+B
(g) B+D
(h) C+A
(i) C+C
(j) C+D
(k) D+A
(l) D+C
(m) D+D
(n) D+B

Question 3
Which of the following products can be made from these matrices ?
$A=\begin{bmatrix} 1 & 2 & -1 \\ 8 & 4 & 7 \\ \end{bmatrix}$
$B=\begin{bmatrix} 1 & 8 \\ 2 & 4 \\ -1 & 7 \\ \end{bmatrix}$

$C= \begin{bmatrix} 1 & -1 & -11 \\ 3 & -6 & 0 \\ \end{bmatrix}$
$D=\begin{bmatrix} 1 & 3 \\ -1 & 1 \\ \end{bmatrix}$
(a) AB
(b) AC
(d) A2
(e) BA
(f) BD
(g) B2
(h) CA
(i) CB
(j) CD
(k) CB
(l) C2
(m) DA
(n) DC
(o) CB
(p) D2
Question 4
$A=\begin{bmatrix} 1 & 2 & -1 \\ 8 & 4 & 7 \\ \end{bmatrix}$
$B= \begin{bmatrix} 1 & 8 \\ 2 & 4 \\ -1 & 7 \\ \end{bmatrix}$
$D= \begin{bmatrix} 1 & 3 \\ -1 & 1 \\ \end{bmatrix}$
Find the following
(a) AB+D
(b) BD
(c)BA+D
(d) D2
(e) DA
(f) ABD

Question 5
Construct A= a2×2 Matrix when the elements are of the forms
(i) $a_{ij}=e^{2ix} sin jx$
$\begin{bmatrix} e_{2x} sin x & e_{2x} sin x \\ e_{4x} sin x & e_{4x} sin 2x \\ \end{bmatrix}$
(ii)
$a_{ij} = 2i- 3 j$
$\begin{bmatrix} -1 & -4 \\ 1 & -2 \\ \end{bmatrix}$

Question 6
If a matrix has 28 elements, what are the possible orders it can have? What if it has 13 elements?

28 × 1, 1 × 28, 4 × 7, 7 × 4, 14 × 2, 2 × 14. If matrix has 13 elements then its order will be either 13 × 1 or 1 × 13.

Question 7
If
$A=\begin{bmatrix} 1 & 2 \\ 4 & 1 \\ \end{bmatrix}$

find A2 + 2A + 7I.

$\begin{bmatrix} 18 & 8 \\ 16 & 18 \\ \end{bmatrix}$

Question 8
If matrix $A = [a_{ij}]_{ 2 \times 2}$, where $a_{ij} = 1$ if i ≠ j
$a_{ij} = 0$ if i = j
then A2 is equal to
(A) I
(B) A
(C) 0
(D) None of these

(A)

Question 9
Find the Matrix C , such that A+B+C is zero matrix
$A=\begin{bmatrix} 2 & 0 & 1 \\ 3 & -1 & 0 \\ \end{bmatrix}$
$B=\begin{bmatrix} 2 & 1 & -1 \\ 0 & 2 & 1 \\ \end{bmatrix}$

$\begin{bmatrix} -4 & -1 & 0 \\ -3 & -1 & -1 \\ \end{bmatrix}$

Question 10
Find the value p and q from the below equation
$2 \begin{bmatrix} p & 5 \\ 17 & q-3 \\ \end{bmatrix} + \begin{bmatrix} 3 & 4 \\ 1 & 2 \\ \end{bmatrix} = \begin{bmatrix} 7 & 14 \\ 15 & 14 \\ \end{bmatrix}$

p =2 and q=9

Question 11
Find the transpose of each of the following matrices:
$A=\begin{bmatrix} 2 & 0 & 1 \\ 3 & -1 & 0 \\ \end{bmatrix}$

$B=\begin{bmatrix} cos \theta & -sin \theta \\ sin \theta & cos \theta \\ \end{bmatrix}$

Question 12
If A is an n × m matrix find the order of AAT and AT A

Question 13
Find the value of p and q
When
$M=\begin{bmatrix} 3 & 1 \\ 7 & 5 \\ \end{bmatrix}$
Such that
M2 + pI= qM

p=q=8

Question 14
$A=\begin{bmatrix} 0 & 3 \\ 2 & -5 \\ \end{bmatrix}$
and
$kA=\begin{bmatrix} 0 & 4a \\ -8 & 5b \\ \end{bmatrix}$
Find the value of a and k

Question 15
if
$f(x)=\begin{bmatrix} cos x & sinx \\ -sin x & cos x \\ \end{bmatrix}$
Then show that
F(A) F(B) = F(A+B)

Question 16
Using elementary row operations , find the inverse of the matrix
$A=\begin{bmatrix} 2 & -1 & 3 \\ -5 & 3 & 1 \\ -3 & 2 & 3 \\ \end{bmatrix}$

Question 17
let
$A=\begin{bmatrix} 2 & 3 \\ 1 & 2 \\ \end{bmatrix}$
and $B=\begin{bmatrix} 4 & -6 \\ -2 & 4 \\ \end{bmatrix}$
Find AB
Solve the following system of equation from it
2x + y=4
3x+ 2y=1

Question 18
Using elementary row operations , find the inverse of the matrix
$A=\begin{bmatrix} 3 & -3 & 4 \\ 2 & -3 & 4 \\ 0 & -1 & 1 \\ \end{bmatrix}$
And hence find the solutions of the equations
3x-3y+4z=21
2x-3y+4z=20
-y + z=5

Question 19
Using elementary row operations , find the inverse of the matrix
$A=\begin{bmatrix} 1 & 1 & 1 \\ 0 & 1 & 3 \\ 1 & -2 & 1 \\ \end{bmatrix}$
And hence find the solutions of the equations
x+y+z=6
y+3z=11
x-2y + z=0

Question 20
Using elementary row operations , find the inverse of the matrix
$A=\begin{bmatrix} 2 & 3 & 1 \\ 2 & 4 & 1 \\ 3 & 7 & 2 \\ \end{bmatrix}$

Question 21
Find the solutions of the equations using matrix method
3x-2y+3z=8
2x +y-z=1
4x-3y + 2z=4

Question 22
Show that for Matrix
$A=\begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & -3 \\ 2 & -1 & 3 \\ \end{bmatrix}$
$A^3 -6A^2+5A+11I=0$
Hence find $A^{-1}$

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