The value of expressions BC , BC T , Bb , b T B . The value of expressions BC , BC T , Bb , b T B are [ − 9 − 5 3 − 1 4 4 ] , Not defined , [ 0 − 8 2 ] , [ 0 − 8 2 ] _ , respectively. Given: The matrices are A = [ 4 − 2 3 − 2 1 6 1 2 2 ] , B = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] , C = [ 0 1 3 2 − 2 0 ] , a = [ 1 − 2 0 ] , b = [ 3 1 − 1 ] . Definition used: “Multiplication of matrices: The product C = AB (in this order) of an m × n matrix A = [ a j k ] times an r × p matrix B = [ b j k ] is defined if and only if r = n and is then the m × p matrix C = [ c j k ] with entries c j k = ∑ l = 1 n a j l b l k = a j 1 b 1 k + a j 2 b 2 k + ⋯ + a j n b n k , where j = 1 , 2 , … , m and k = 1 , 2 , … , p .” Transpose of a matrix: “The transpose of an m × n matrix A = a j k is the n × m matrix A T that has the first row of A as its first column , the second row of A as its second column , and so on. Thus the transpose of A is A T = [ a k j ] .” Calculation: The product of the matrix B and C is obtained as follows. BC = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] [ 0 1 3 2 − 2 0 ] = [ 1 ( 0 ) + ( − 3 ) 3 + 0 ( − 2 ) 1 ( 1 ) + ( − 3 ) 2 + 0 ( 0 ) − 3 ( 0 ) + 1 ( 3 ) + 0 ( − 2 ) − 3 ( 1 ) + 1 ( 2 ) + 0 ( 0 ) 0 ( 0 ) + 0 ( 3 ) + ( − 2 ) ( − 2 ) 0 ( 1 ) + 0 ( 2 ) + ( − 2 ) ( 0 ) ] = [ − 9 − 5 3 − 1 4 4 0 ] Use the definition of transpose and obtain the value of C T as C T = [ 0 3 − 2 1 2 0 ] . Observe that the order of the matrix B is 3 × 3 , whereas the order of the matrix C T is 2 × 3 . Since the number of columns of matrix B is not equal to the number of rows of matrix C T , the matrix multiplication BC T is not defined. Find the product of Bb . Bb = [ 1 − 3 0 − 3 1 0 0 0 − 2 ] [ 3 1 − 1 ] = [ 1 ( 3 ) + ( − 3 ) ( 1 ) + 0 ( − 1 ) ( − 3 ) ( 3 ) + 1 ( 1 ) + 0 ( − 1 ) 0 ( 3 ) + 0 ( 1 ) + ( − 2 ) ( − 1 ) ] = [ 0 − 8 2 ] Observe that the matrix B is symmetric and hence, B = B T . Since ( AB ) T = B T A T , the value of b T B is obtained as follows. b T B = b T B T = ( Bb ) T = [ 0 − 8 2 ] T = [ 0 − 8 2 ] Thus, the value of expressions BC , BC T , Bb , b T B are [ − 9 − 5 3 − 1 4 0 ] , Not defined , [ 0 − 8 2 ] , [ 0 − 8 2 ] _ , respectively.

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