Matrix Algebra

Question 1: Consider the matrices A = [ 2 3 4 1 ] [ 2 4 ​ 3 1 ​ ] and B = [ − 1 2 0 3 ] [ −1 0 ​ 2 3 ​ ]. Calculate the product  AB if possible. Solution 1: To calculate the product  AB, we multiply the elements of each row of matrix A with the corresponding elements of each column of matrix B and then sum the products. Here's how the calculation is done: = [ 2 3 4 1 ] [ − 1 2 0 3 ] = [ ( 2 ⋅ − 1 + 3 ⋅ 0 ) ( 2 ⋅ 2 + 3 ⋅ 3 ) ( 4 ⋅ − 1 + 1 ⋅ 0 ) ( 4 ⋅ 2 + 1 ⋅ 3 ) ] = [ − 2 13 − 4 11 ] AB=[ 2 4 ​ 3 1 ​ ][ −1 0 ​ 2 3 ​ ]=[ (2⋅−1+3⋅0) (4⋅−1+1⋅0) ​ (2⋅2+3⋅3) (4⋅2+1⋅3) ​ ]=[ −2 −4 ​ 13 11 ​ ] Question 2: Find the transpose of the matrix C = [ 5 − 2 7 0 3 1 ] [ 5 0 ​ −2 3 ​ 7 1 ​ ]. Solution 2: To find the transpose of matrix C, we simply swap its rows with columns. Here's the calculation: Transpose of C = [ 5 0 − 2 3 7 1 ] ⎣ ⎡ ​ 5 −2 7 ​ 0 3 1 ​ ⎦ ⎤ ​ Question 3: If matrix D = [ 2 1 − 3 0 ] [ 2 −3 ​ 1 0 ​ ], calculate its determinant. Solution 3: To calculate the determinant of matrix D, we use the formula  det(D)=ad−bc, where  a, b, c, and d are the elements of the matrix. Here's the calculation: (  ) = ( 2 ⋅ 0 ) − ( − 3 ⋅ 1 ) = 0 + 3 = 3 det(D)=(2⋅0)−(−3⋅1)=0+3=3