Q. 175.0( 1 Vote )

# Given and . Is (AB)’ = B’A’?

We have two matrices A and B, such that

We need to verify whether (AB)’ = B’A’.

Let us understand what a transpose is.

In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT.

Take L.H.S = (AB)’

So, let us compute AB.

Multiply 1st row of matrix A by matching members of 1st column of matrix B, then sum them up.

(2, 4, 0)(1, 2, 1) = (2 × 1) + (4 × 2) + (0 × 1)

(2, 4, 0)(1, 2, 1) = 2 + 8 + 0

(2, 4, 0)(1, 2, 1) = 10

Multiply 1st row of matrix A by matching members of 2nd column of matrix B, then sum them up.

(2, 4, 0)(4, 8, 3) = (2 × 4) + (4 × 8) + (0 × 3)

(2, 4, 0)(4, 8, 3) = 8 + 32 + 0

(2, 4, 0)(4, 8, 3) = 40

Similarly, let us do it for the rest of the elements.

So,

Now, for transpose of AB, rows will become columns.

Now, take R.H.S = B’A’

If

Then, if (1, 4) are the elements of 1st row, it will become elements of 1st column, and so on.

Also,

Then, if (2, 4, 0) are the elements of 1st row, it will become elements of 1st column, and so on.

Now, multiply B’A’.

Multiply 1st row of matrix B’ by matching members of 1st column of matrix A’, then sum them up.

(1, 2, 1)(2, 4, 0) = (1 × 2) + (2 × 4) + (1 × 0)

(1, 2, 1)(2, 4, 0) = 2 + 8 + 0

(1, 2, 1)(2, 4, 0) = 10

Multiply 1st row of matrix B’ by matching members of 2nd column of matrix A’, then sum them up.

(1, 2, 1)(3, 9, 6) = (1 × 3) + (2 × 9) + (1 × 6)

(1, 2, 1)(3, 9, 6) = 3 + 18 + 6

(1, 2, 1)(3, 9, 6) = 27

Similarly, filling up for the rest of the elements.

L.H.S = R.H.S

Thus, (AB)’ = B’A’.

Rate this question :

How useful is this solution?
We strive to provide quality solutions. Please rate us to serve you better.
Related Videos
Determinants of Matrices of different order59 mins
Triangular Matrices & operations on matrices58 mins
Determining a determinant63 mins
Types of Matrices & Properties51 mins
Interactive Quiz on Matrices and Determinants41 mins
Test Yourself, Properties of Determinants30 mins
Interactive Quiz on Matrices & Determinants48 mins
Interactive Quiz on Properties of Determinants43 mins
Lecture on Product of Determinants58 mins