The value of expressions ABC , ABa , ABb , Ca T . The value of expressions ABC , ABa , ABb , Ca T are [ − 30 − 18 45 9 5 − 7 ] , Not defined , [ 22 4 − 12 ] , Not defined _ , 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 A and B is given by, AB = [ 4 − 2 3 − 2 1 6 1 2 2 ] [ 1 − 3 0 − 3 1 0 0 0 − 2 ] = [ 4 ( 1 ) + ( − 2 ) ( − 3 ) + 3 ( 0 ) 4 ( − 3 ) + ( − 2 ) ( 1 ) + 3 ( 0 ) 4 ( 0 ) + ( − 2 ) ( 0 ) + 3 ( − 2 ) − 2 ( 1 ) + 1 ( − 3 ) + 6 ( 0 ) − 2 ( − 3 ) + 1 ( 1 ) + 6 ( 0 ) − 2 ( 0 ) + 1 ( 0 ) + 6 ( − 2 ) 1 ( 1 ) + 2 ( − 3 ) + 2 ( 0 ) 1 ( − 3 ) + 2 ( 1 ) + 2 ( 0 ) 1 ( 0 ) + 2 ( 0 ) + 2 ( − 2 ) ] = [ 10 − 14 − 6 − 5 7 − 12 − 5 − 1 − 4 ] Now, multiply the above value with matrix C . ABC = [ 10 − 14 − 6 − 5 7 − 12 − 5 − 1 − 4 ] [ 0 1 3 2 − 2 0 ] = [ 10 ( 0 ) + ( − 14 ) ( 3 ) + ( − 6 ) ( − 2 ) 10 ( 1 ) + ( − 14 ) ( 2 ) + ( − 6 ) ( 0 ) − 5 ( 0 ) + 7 ( 3 ) + ( − 12 ) ( − 2 ) − 5 ( 1 ) + 7 ( 2 ) + ( − 12 ) ( 0 ) − 5 ( 0 ) + ( − 1 ) ( 3 ) + ( − 4 ) ( − 2 ) − 5 ( 1 ) + ( − 1 ) ( 2 ) + ( − 4 ) ( 0 ) ] = [ − 30 − 18 45 9 5 − 7 ] Observe that the order of the matrix AB is 3 × 3 , whereas the order of the matrix a is 1 × 3 . Since the number of columns of matrix AB is not equal to the number of rows of matrix a , the matrix multiplication ABa is not defined. Find the product of A Bb , where the matrix AB is given by [ 10 − 14 − 6 − 5 7 − 12 − 5 − 1 − 4 ] . ABb = [ 10 − 14 − 6 − 5 7 − 12 − 5 − 1 − 4 ] [ 3 1 − 1 ] = [ 10 ( 3 ) + ( − 14 ) ( 1 ) + ( − 6 ) ( − 1 ) ( − 5 ) ( 3 ) + 7 ( 1 ) + ( − 12 ) ( − 1 ) ( − 5 ) ( 3 ) + ( − 1 ) ( 1 ) + ( − 4 ) ( − 1 ) ] = [ 22 4 − 12 ] Use the definition of transpose mentioned above and obtain the transpose of matrix a as, a T = [ 1 − 2 0 ] . Here, the order of the matrix C is 3 × 2 , whereas the order of the matrix a T is 3 × 1 . Since the number of columns of matrix C is not equal to the number of rows of matrix a T , the matrix multiplication Ca T is not defined. Thus, the value of expressions ABC , ABa , ABb , Ca T are [ − 30 − 18 45 9 5 − 7 ] , Not defined , [ 22 4 − 12 ] , Not defined _ , respectively.

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