Answer:
1. Answer: (b) 4
Explanation: As 6 → 1 × 6, 2 × 3, 3 × 2, 6 × 1.
2. Answer: (d) 64
Explanation: The order of the matrix =2×3
The number of elements =2×3=6
Each place can have either 0 or 1.
Each place can be filled in 2 ways.
Thus, the number of possible matrices
=26 =64
3. Answer: (a) k = 3, p = n
Explanation: In this, order of P=p×k
Order of W=n×3
Order of Y=3×k
Thus, order of PY=p×k, when k=3.
And the order of WY=p×k, where p=n
4. Answer: (a) I
Explanation: (a), as (I + A)2 -3A
= I2 + IA + AI + A2 – 3A
= I + A + A + A – 3A=I
5. Answer: (b) m × n
Explanation: If A is matrix of order m×n and B is a matrix of order n×p, then the order of AB is m×p.
6. Answer: (a) rectangular matrix
Explanation: If we exchange the role of the row and column of a matrix then we transpose it. This operation is called transposition and the matrix is called transposition matrix.
If we take a rectangular matrix and transpose it then we get a rectangular matrix.
7. Answer: (d) diagonal matrix
Explanation: A diagonal matrix is a matrix in which all of the elements not on the diagonal of a square matrix are 0.
8. Answer: (c) m = n
Explanation: MatrixA=[aij]m×n square matrix
Number of columns = number of rows
n=m
9. Answer: (b) AB = BA = I
Explanation: We know that if A is a square of order m, and if there exists another square matrix B of the same order m, such that AB=I, then B is said to be the inverse of A.
In this case, it is clear that A is the inverse of B.
Thus , matrices A and B will be inverses of each other only if AB=BA=I.
10. Answer: (a) all zeroes
Explaination: As in skew symmetric matrix, aij = -aji
⇒ aii = – aii
⇒ 2aii = 0
⇒ aii = 0, i.e. diagonal elements are zeroes.
11. Answer: c
Explanation: If matrix A is symmetric
AT=A
If matrix A is skew-symmetric
AT=−A
Also, diagonal elements are zero
Now, it is given that a matrix A is both symmetric as well as skew-symmetric
∴A=AT=−A
which is only possible if A is zero matrix
A=[(0,0)(0,0)]=AT=−A
12. Answer: (d) 512
Explanation: The number of possible entries of 3×3 matrix is 9.
Every entry has two choices, 0 or 1
Thus, total number of choices is
2×2×2×2×2×2×2×2×2 = 29 = 512
13. Answer: (d) C
14. Answer: (d) B - 1 A - 1
15. Answer: (b) a skew-symmetric matrix
Explanation: A and B are two symmetric matrices.
A = A’ and B = B’
(AB – BA)’ = (AB)’ – (BA)’ = B’A’ – A’B’
= BA – AB = – (AB – BA)
(AB – BA)’ = – (AB – BA)
AB – BA is a skew-symmetric matrix.
16. Answer: (b) skew-symmetric matrix
Explanation: Let's say that B = A - A', where A is a square matrix. We know that if the transpose is equal to the negative of a matrix, then it is a skew-symmetric matrix.
17. Answer: (b) A – AT
18. Answer: (b) symmetric matrix
Explanation: Let A be any matrix.
Also let B=AAT.
Now BT=(AAT)T=(AT)TAT=AAT=B. [ Since (AT)T=A]
So AAT is a symmetric matrix.
19. Answer: (c) 4 elements
Explanation: Its order is 3*4 which means it has three rows and four columns So every row has four elements.
20. Answer: (b) 2 × n
Explanation: Given, order of X = 2×n
and order of Z = 2×p
Therefore, n=p
Hence the order of 7X−5Z = 2×n.
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