𝑥1
𝑥
1. If for any given vector x= [𝑥2 ] , the product Ax is the column vector given below, find A.
3
𝑥4
(i)
2𝑥1 − 𝑥3 − 𝑥4
[ −2𝑥1 + 𝑥2 ]
𝑥2 + 𝑥4
(ii)
𝑥1 + 3𝑥4
𝑥2 − 𝑥4
𝑥3 + 𝑥4
𝑥4
[𝑥3 − 2𝑥1 ]
2. Verify (ABC)T = C TB TA T directly for
5
A = [0
1
−2
0
1 ] ,B=[ ]
7
3
, C =[3,1,2,9]
3. Prove that the product AB need not be symmetric , even if A and B are both symmetric
and of the same order.
4. Let A = [4 1 2] , B = [1 −4 2] , evaluate (BAT)T
0
5
7
8
1
4
5. Find the rank of a matrix
2
𝐴 = [2
0
2
1
4
3
1
−3
−2
1
−3
1 2 3
5 −1 0
𝐴 = 6 −1 2
7 −1 0
[8 −1 −3
4
5]
3
−2
4
5
8
7
5]
6. Check weather the matrices are orthogonal or no?
1
0
𝐴=[
0
0
0
0
1
0
0
0
1/√2 1/√2
]
0
0
−1/√2 1/√2
1/√2
1/√3 1/√6
𝐴 = [1/√2 −1/√3 −1/√6]
0
1/√3 −2/√6
7. A) Prove that if Q is orthogonal ,then so is QT
B) Prove that if Q is orthogonal, then so is Q-1
8. For a given matrix 𝐴 = [
6
4 2 −5
] and 𝐵 = [ 0 ] Prove that (AB)T=BTAT
0 1 3
−1
2
9. Find the determinant of 𝐴 = |0
0
1
0
3
4
2
1
1
5
3
0
−1|
0
6