Question

Matrix Algebra

3. indicated sums and products are defined. Mark each statement True of False. Justify each The following questions concern arbitrary matrices A, B, and C for which the answer (a) If A and B are 2 x 2 matrices with columns ai, a2 and bi, b2, respectively, thern 101 a2b2] (b) Each column of AB is a linear combination of the columns of B using weights from the corresponding column of A. (c) AB + AC-A(B +C (d) AT BT (AB)T e) The transpose of a product of matrices equals the product of their transposes in the same order.

Answer 1

(a) False

A = [a₁ a₂] & B = [b₁ b₂]

So, AB   $\neq$ [a₁b₁ a₂b₂]

(b) True

By the very definition of the product of 2 matrices.

(c) True

By distributive law (left) of product of matrices over addition of matrices.

(d) True

By definition & properties of the transpose of the addition of 2 matrices.

(e) False

Let us check the case for 2 matrices.

We have, (AB)^T = (B^TA^T)

So, the transpose of a product of matrices equals the product of their transposes in the reverse order.