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   0 12 ) = ( 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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