Multiplication of Two Matrices (Examples)


Example 1:
Find the product of the following pairs of matrices.
(a) ( 1      5      2 )( 2 4 3 ) (b) ( 2 8 3 1 )( 1 0 4 2 ) (c) ( 3 5 )( 2      6 ) (d) ( 0 4 1 3 )( 7 2 ) (e) ( 7      4 )( 2 0 1 3 )

Solution:
(a)  ( 1      5      2 )( 2 4 3 ) Matrices analysis 1×3 and 3×1             =1×1 matrix =( 1×2  5×4  2×3 ) =( 2+20+6 ) =( 28 )

(b)

( 2 8 3 1 )( 1 0 4 2 ) Matrices analysis 2×2 and 2×2               =2×2 matrix =( 2×1+8×4   2×0+8×2 3×1+1×4   3×0+1×2 ) =( 34 16 1 2 )

(c)

( 3 5 ) ( 2      6 ) Matrices analysis 2 × 1  and 1 × 2               = 2 × 2  matrix = ( 3 × 2    3 × 6 5 × 2      5 × 6 ) = ( 6 18 10 30 )

(d)

( 0 4 1 3 )( 7 2 ) Matrices analysis 2×2 and 2×1              =2×1 matrix =( 0×7+4×2 1×7+3×2 ) =( 8 13 )

(e)

( 7      4 )( 2 0 1 3 ) Matrices analysis 1×2 and 2×2               =1×2 matrix =( 7×2+( 4×1 )         7×0+( 4×3 ) ) =( 14+4       012 ) =( 10     12 )



Example 2:
Find the values of m and n in each of the following matrix equations.
( a ) ( 3 m ) ( 1       n ) = ( 3 12 2 8 ) ( b ) ( m 2 3 1 ) ( 2 n ) = ( 12 4 + 2 n ) ( c ) ( m 3 1 1 ) ( 1 2 4 n ) = ( 14 11 5 3 )
 
Solution:
(a) ( 3 m ) ( 1   n ) = ( 3 12 2 8 ) ( 3 3 n m m n ) = ( 3 12 2 8 ) m = 2 ,   3 n = 12 n = 4

(b) ( m 2 3 1 ) ( 2 n ) = ( 12 4 + 2 n ) ( 2 m + 2 n 6 + n ) = ( 12 4 + 2 n ) 6 + n = 4 + 2 n n = 10 2 m + 2 n = 12 2 m + 2 ( 10 ) = 12 2 m 20 = 12 2 m = 32 m = 16

(c) ( m 3 1 1 ) ( 1 2 4 n ) = ( 14 11 5 3 ) ( m + ( 12 ) 2 m + ( 3 n ) 1 + 4 2 + n ) = ( 14 11 5 3 ) m 12 = 14 m = 2 m = 2 2 + n = 3 n = 5


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