DETERMINANTS & MATRICES (UNIT: 1)
OBJECTIVE TYPE MULTIPLE CHOI𝑪E
0 −1 2
1. The matrix ( 1 0 −3) is (a) symmetric (b) skew- symmetric (c) a diagonal (d) none
−2 3 0
1 2 .3
2. The matrix (0 4 5 ) is (a) lower triangular (b) upper triangular (c) a diagonal (d) none
0 0 6
0 −1 2
3. The matrix ( 1 0 −3) is (a) symmetric (b) skew- symmetric (c) a diagonal (d) none
−2 3 0
1 −1 2
4. The matrix (−1 2 4) is (a) lower triangular (b) upper triangular (c)a diagonal (d) none
2 4 3
1 0 0
5. The matrix (5 4 0) is (a) lower triangular (b) upper triangular (c) a diagonal (d) none
3 2 6
1 0 0
6. The matrix (0 −1 0) is (a) unit matrix (b) scalar matrix (c) a diagonal matrix (d) none
0 0 1
5 0 0
7. The matrix (0 1 0) is (a) unit matrix (b) scalar matrix (c) a diagonal matrix (d) none
0 0 1
3 0 0
8. The matrix (0 3 0) is (a) unit matrix (b) scalar matrix (c) a diagonal matrix (d) none
0 0 3
2𝑥 − 𝑦 5 6 5
9. If ( ) = ( ) then value of x & y = (a) (2,-2) (b) (-2,2) (c) (3,0) (d) none
3 𝑦 3 −2
𝑥 𝑦 4 3 2 6
10. . If 3× ( ) = ( )+( ) then value of x , y , z & t = (a) (2,3,4,3) (b) (2,3,3,4)
𝑧 𝑡 5 0 4 12
( c) ( 2,0,3,4) (d) none of these
𝑥 + 4 𝑥 + 3𝑦 0 8
11. . If ( ) = ( ) then value of x , y , z & t = (a) (4,-4,1,-2) (b) (-4,4,1,2)
𝑧 + 1 3𝑡 − 4 2 1
( c) ( -4,4,-1,-2) (d) none of these
𝑥 2 3 4 3 8
12. If 2 ( ) +( )= ( ) then value of x & y = (a) (0,0) (b) (1,0) (c) (1,1)
7 𝑦+5 1 2 15 14
(d) none of these
13. If A & B be two square matrices then which one is correct (a) det (AB) = det(A) .det(B)
(b) det(A+B) = det(A) + det(B) (c) det(A-B) = det(A) - det(B (c) none of these
1 0 0
14. 𝐼3 = (0 1 0) then 𝐼3 2 + 2 𝐼3 (a) 𝐼3 (b) 2 𝐼3 (c) 3𝐼3 (d) none of these
0 0 1
1 0
15. 𝐼2 = ( ) then 𝐼2 5 + 𝐼2 (a) 8 𝐼2 (b) 5 𝐼2 (𝑐) 3 𝐼2 (𝑑)8 𝐼2
0 1
16. A square matrix is singular if (a) det A = 0 (b) det A ≠ 0 (c) detA = 1 (d) det A ≠ 1
2 𝑎
17. Find the value of a if A = ( ) is singular (a) 4 (b) 0 (c) -4 (d) 12
3 6
18. If A and B are two square matrices such that 𝐴2 = A , 𝐵 2 = B then
(a) ( 𝐴𝐵)2 = 1 (b) ( 𝐴𝐵)2 = AB (c) ( 𝐴𝐵)2 = BA (d) None of these
19. Given that matrix A has size 3 × 4 . Then 𝐴𝑇 B and B𝐴𝑇 are possible when B has size
(a) 4× 4 (b) 4×3 (c) 3×4 (d) 3×3
20. . Given that matrix A has size 4 × 5. Then 𝐴𝑇 B and B𝐴𝑇 are possible when B has size
(a) 4× 5 (b) 5×5 (c) 5×4 (d) 4×4
1 0 0 1
1 2 0 2 0
21. If A = = (2 2) Then 𝐴𝑇 is (a) ( ) (b) = ( ) (c) (2 2) (d) None
0 2 4 4 1
0 4 4 0
1 2 2 1
22. If A = ( ) and B = ( ) , find 2A – B.
3 4 4 3
𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
23. If A = ( ) and B = ( ) , find AB .
1 0 𝑐𝑜𝑠𝛼 𝑠𝑖𝑛𝛼
𝑠𝑖𝑛𝛼 𝑐𝑜𝑠𝛼
24. If A = ( ) find 𝐴2 .
𝑐𝑜𝑠𝛼 −𝑠𝑖𝑛𝛼
5 0
25. Find the adjoint of the matrix and A = ( )
1 −2
26. If A is a symmetric matrix prove that 𝐴 𝐴𝑇 is symmetric matrix
27. If A is a skew symmetric matrix prove that 𝐴 𝐴𝑇 is symmetric matrix.
28 If A is square matrix then prove that (i) 𝐴 + 𝐴𝑇 is symmetric matrix
(ii) ) 𝐴 − 𝐴𝑇 is skew symmetric matrix
SUBJECTIVE
−1 2 −2
1
1.If A = 3 (−2 1 2 ) , Show that 𝐴𝐴𝑇 = Ι
2 2 1
1 2 3
2. If A = (3 4 5) then show that A + 𝐴𝑇 is a symmetric matrix and A – 𝐴𝑇 skew - symmetric.
5 6 7
1 2 3
3. Express A= (3 4 5) as sum of two matrices such that one of them is symmetric and
5 6 7
other is skew-symmetric
1 2 3 1 2 3
4. If A = (6 7 8 ) and B = (3 4 2 ) , then show that (𝐴𝐵)𝑇 = 𝐵 𝑇 𝐴𝑇 and
6 −3 4 5 6 1
(𝐴 + 𝐵)𝑇 = 𝐴𝑇 + 𝐵 𝑇
1 2 2
5. Show that A = (2 1 2) , satisfies the equation 𝐴2 -4A - 5Ι = 0 and hence find 𝐴− 1 .
2 2 1
1 2 3 1 2 3
6. Find Adj, If A = (2 −4 5 ) 7. If A = (2 −4 5 ) then find 𝐴− 1
6 1 0 6 1 0
0 1 3
8. If A = (1 2 3 ) Find the inverse of A
3 1 1
0 −1 0 −1 1 −1
−1
9. If A = (1 1 4 ) and show that A𝐴 = Ι 10.If A = ( 3 −3 3 ) ,show that A is
0 −1 1 5 −5 5
nilpotent.
2 −1
11.If A = ( ) , then show that 𝐴2 -4A - 7Ι = 0 and hence find 𝐴− 1 .
3 2
𝑎 𝑏 𝑐 0 0 1
12. If A = (𝑑 𝑐 −4) and B = (0 1 0 ) , if AB = BA , find a, b, c, d .
5 −6 7 1 0 0
−1 1 −1
13 If A = ( 3 −3 3 ) , show that A is idempotent matrix
5 −5 5
0 −1 0
14. If A = (1 1 4 ) , show that A is involutary matrix.
0 −1 1
