Determine whether the product of the matrices is defined in each case. If so, state the

order of the product.

(i) AB, where A = [a_ij]_4x3, B = [b_ij]_3x2

(ii)PQ, where P = [p_ij]_4x3, Q = [q_ij]_4x3

(iii)MN, where M = [m_ij]_3x1, N = [n_ij]_1x5

(iv) RS, where R = [r_ij]_2x2, S = [s_ij]_2x2

(i) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here A[a_ij]_{4 x 3} and B = [b_ij]_3x2

⇒ Number of columns in A = 3

⇒ Number of rows in B = 3

Thus the product is defined and the order if product is

Number of rows in A × Number of columns in B

∴ AB = 4 × 3

(ii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here P[p_ij]_{4 x 3} and Q = [q_ij]_4x3

⇒ Number of columns in P = 3

⇒ Number of rows in Q = 4

Thus the product is not defined.

(iii) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here M[m_ij]_{3 x 1} and N = [n_ij]_1x5

⇒ Number of columns in M = 1

⇒ Number of rows in N = 1

Thus the product is defined and the order if product is

Number of rows in M × Number of columns in N

∴ MN = 3 × 5

(iv) The multiplication of 2 matrices is possible if number of columns in first matrix is equal to number of rows in second.

⇒ Here R[r_ij]_{2 x 2} and S = [s_ij]_2x2

⇒ Number of columns in R = 2

⇒ Number of rows in S = 2

Thus the product is defined and the order if product is

Number of rows in R × Number of columns in S

∴ RS = 2 × 2