four parts to this one question a)show that[a 0] [0 b] 2x2 matrix is invertible
ID: 3076262 • Letter: F
Question
four parts to this one question
a)show that[a 0]
[0 b] 2x2 matrix is invertible if and only if a#0 and b#0. Describe the inverse.
b) show that a diagonal matrix is invertible if and only if all the main diagonal entries are non zero.Describe the inverse.
c) If A and B are square matrices,show that 1)the block matrix [A 0]
[0 B] is invertible if and only if A and B are both invertible and 2)[A0] [A-1 0]
[0B]inverse = [0 B-1]
d) use part c to find the inverses of 1)[100] 2)[310]
[02-1] [520]
[01-1] [00-1]
3)[2100] 4)[3400]
[1100] [2300]
[001-1] [0013]
[001-2] [000-1]
Explanation / Answer
a)inverse exists if matrix A is non- singular=>|A|#0 |A|=ab#0=>a#0 and b#0 b)similarly in case of diagonal matrix |diagonal matrix|=product of diagonal elements which is not equal to 0 implies all the main diagonal entries are non zero c)since A and B are square matrices |A|#0 and |B|#0 hence [A 0] [0 B] is invertible if and only if A and B are both invertible
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