Question

3.Compute the indicated products,()212 3 4()21 -211 2 3]3 2 3 111) D2 1(v) 3 22 3 413 54 5 6 3 0 53-1 3(iv) 3 4 50 2 41V(vi)-1 0 2-」131

Answer 1

(i)

The given matrices are,

[ a b −b a ][ a −b b a ]

The product of given two matrices is,

[ a b −b a ][ a −b b a ]=[ a( a )+b( b ) a( −b )+b( a ) −b( a )+a( b ) −b( −b )+a( a ) ] =[ a 2 + b 2 −ab+ab −ab+ab b 2 + a 2 ] =[ a 2 + b 2 0 0 a 2 + b 2 ]

Thus, the product of the given matrices is [ a 2 + b 2 0 0 a 2 + b 2 ].

(ii)

The given matrices are,

[ 1 2 3 ][ 2 3 4 ]

The product of given two matrices is,

[ 1 2 3 ][ 2 3 4 ]=[ 1( 2 ) 1( 3 ) 1( 4 ) 2( 2 ) 2( 3 ) 2( 4 ) 3( 2 ) 3( 3 ) 3( 4 ) ] =[ 2 3 4 4 6 8 6 9 12 ]

Thus, the product of two matrices is [ 2 3 4 4 6 8 6 9 12 ].

(iii)

The given matrices are,

[ 1 −2 2 3 ][ 1 2 3 2 3 1 ]

The product of two matrices is,

[ 1 −2 2 3 ][ 1 2 3 2 3 1 ]=[ 1( 1 )−2( 2 ) 1( 2 )−2( 3 ) 1( 3 )−2( 1 ) 2( 1 )+3( 2 ) 2( 2 )+3( 3 ) 2( 3 )+3( 1 ) ] =[ 1−4 2−6 3−2 2+6 4+9 6+3 ] =[ −3 −4 1 8 13 9 ]

Thus, the product of two matrices is [ −3 −4 1 8 13 9 ].

(iv)

The given matrices are,

[ 2 3 4 3 4 5 4 5 6 ][ 1 −3 5 0 2 4 3 0 5 ]

The product of two matrices is,

[ 2 3 4 3 4 5 4 5 6 ][ 1 −3 5 0 2 4 3 0 5 ]=[ 2( 1 )+3( 0 )+4( 3 ) 2( −3 )+3( 2 )+4( 0 ) 2( 5 )+3( 4 )+4( 5 ) 3( 1 )+4( 0 )+5( 3 ) 3( −3 )+4( 2 )+5( 0 ) 3( 5 )+4( 4 )+5( 5 ) 4( 1 )+5( 0 )+6( 3 ) 4( −3 )+5( 2 )+6( 0 ) 4( 5 )+5( 4 )+6( 5 ) ] =[ 2+0+12 −6+6+0 10+12+20 3+0+15 −9+8+0 15+16+25 4+0+18 −12+10+0 20+20+30 ] =[ 14 0 42 18 −1 56 22 −2 70 ]

Thus, the product of two matrices is [ 14 0 42 18 −1 56 22 −2 70 ].

(v)

The given matrices is,

[ 2 1 3 2 −1 1 ][ 1 0 1 −1 2 1 ]

The product of two matrices is,

[ 2 1 3 2 −1 1 ][ 1 0 1 −1 2 1 ]=[ 2( 1 )+1( −1 ) 2( 0 )+1( 2 ) 2( 1 )+1( 1 ) 3( 1 )+2( −1 ) 3( 0 )+2( 2 ) 3( 1 )+2( 1 ) −1( 1 )+1( −1 ) −1( 0 )+1( 2 ) −1( 1 )+1( 1 ) ] =[ 2−1 0+2 2+1 3−2 0+4 3+2 −1−1 0+2 −1+1 ] =[ 1 2 3 1 4 5 −2 2 0 ]

Thus, the product of two matrices is [ 1 2 3 1 4 5 −2 2 0 ].

(vi)

The given matrices are,

[ 3 −1 3 −1 0 2 ][ 2 −3 1 0 3 1 ]

The product of two matrices is,

[ 3 −1 3 −1 0 2 ][ 2 −3 1 0 3 1 ]=[ 3( 2 )−1( 1 )+3( 3 ) 3( −3 )−1( 0 )+3( 1 ) −1( 2 )+0( 1 )+2( 3 ) −1( −3 )+0( 0 )+2( 1 ) ] =[ 6−1+9 −9−0+3 −2+0+6 3+0+2 ] =[ 14 −6 4 5 ]

Thus, the product of two matrices is [ 14 −6 4 5 ].