True or False Questions The ith row of the matrix product AB can be computed by

Question

True or False Questions The ith row of the matrix product AB can be computed by multiplying A with the ith row vector of the matrix B. 1) 2) IfAB+BA+Cis defined then A, B and Care square matrices of the same size. 3) For any matrix A, and any positive integer n, we can always compute A". 4) If A and B are nxn matrices then (A+B) A2+B*+2AB 5) (kA+B)-1=-A-1+8-1 6) (A-In) (As + A4 + A3 + Az + A + I 7) If AB is not invertible then either A is not invertible or B is not invertible 8) IfA and B are symmetric matrices of the same size then AB is also a symmetric n) = A6-In for any nxn matrix A. matrix 9) If A is an nxn matrix and B is a symmetric matrix of the same size then ATBA is also a symmetric matrix. 10) It is possible to find a matrix such that A1-A.

Explanation / Answer

1. False

ith row vector of AB is obtained by multiplying matrix A with ith column of B.

2. True

If AB and BA both are defined and their addition is defined, both should have same order. that order must be equal to C.

3. True

We can always compute An

4. False

(A+B)2= A2+B2+AB+BA

5. False

(kA+B)-1is not equal to the above, since the formula for (a+b)-1 is quite complex and not the above equation

6. False

(A-In)(A5+A4+A3+A2+A-In) = A6+A5+A4+A3+A2-A-(A5+A4+A3+A2+A-In) =A6-2A+In

7. True

Det(AB) = DetA.DetB

Given Det(AB) = 0

Det(A)Det(B)=0

Either A or B has determinant 0 hence not invertible.

8. False

If A and B are symmetric, then AB is also symmetric only if AB commutate

9. True

(ATBA)T=(BA)T(AT)T

(ATBA)T=(ATBT)A

(ATBA)T=(ATB)A [since B is symmetric]

10. Yes, It is possible, check for

A = ( 0 -1

1 0)

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