## PRACTICE QUESTIONS ON TRANSPOSE OF MATRIX

Transpose of a Matrix :

The transpose of a matrix is obtained by interchanging rows and columns of A and is denoted by AT.

More precisely, if [aij] with order m x n, then AT  =   [bij] with order n x m,  where bij  =  aji so that the (i, j)th entry of AT is  aji

We state a few basic results on transpose whose proofs are straight forward.

For any two matrices A and B of suitable orders, we have

(i) (AT )T = A

(ii) (kA)T = kAT (where k is any scalar)

(iii) (A+ B)T = AT + BT

(iv) (AB)T = BT AT (reversal law on transpose)

Question 1 :

verify the following (i) (A+ B)T = AT + BT = BT + AT

(ii) (A− B)T = AT − BT

(iii) (BT )T = B .

In order to find the value of (A + B)T , first let us find the value of A. For that we have to find the transpose of matrix A.

By finding transpose for the transposed matrix, we get the original matrix.

To find the value of AT + BT, we have to add those matrices.

To find the value of BT+ AT, we have to add those matrices.

Hence proved.

(ii) (A− B)T = AT − BT

L.H.S

First let us subtract the matrix B from A.Now by interchanging rows and columns, we get (A - B)T

R.H.S

Now we have to subtract BT from AT

Question 2 :

If A is a 3 × 4 matrix and B is a matrix such that both AT B and BAT are defined, what is the order of the matrix B?

Solution :

If A is a matrix of order 3 x 4, then the order of the matrix AT will be 4 x 3. Since AT B is defined, matrix B will have the order 3 x 2 or 3 x 1.

If we take the order 3 x 2 for the matrix B, we cannot find the value of BAT

If the product of two matrices is defined , then the number of columns in the first matrix will be equal to the number of rows in the second matrix.

Hence the required order must be 3 x 4.

So, AT B will have the order 4 x 4 and the matrix BAT will have the order 3 x 3.

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