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

Solution:
(a)( 3 m )( 1   4 )=( 3 12 2 n ) ( 3 12 m 4m )=( 3 12 2 n ) m=2,  4m=n 4(2)=n n=8 (b)( m 2 3 1 )( 2 n )=( 12 4+2n ) ( 2m+2n 6+n )=( 12 4+2n ) 6+n=4+2n n=10 2m+2n=12 2m+2(10)=12 2m20=12 2m=32 m=16 (c)( m 3 1 1 )( 1 2 4 n )=( 14 11 5 3 ) ( m+(12) 2m+(3n) 1+4 2+n )=( 14 11 5 3 ) m12=14 m=2 m=2 2+n=3 n=5