Given the matrices M = [[-3, 4], [5, 6]] and N = [[1, 0], [-5, -7]], find MN and M^T + N^T.

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Asked on April 1, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-04-01T19:51:56.943Z

Given the matrices: M = -3 & 4 \\ 5 & 6 N = 1 & 0 \\ -5 & -7 i) Find MN. Step 1: Multiply the matrices M and N. To multiply two matrices, we multiply the rows of the first matrix by the columns of the second matrix. MN = (-3)(1) + (4)(-5) & (-3)(0) + (4)(-7) \\ (5)(1) + (6)(-5) & (5)(0) + (6)(-7) Step 2: Perform the multiplications and additions. MN = -3 - 20 & 0 - 28 \\ 5 - 30 & 0 - 42 MN = -23 & -28 \\ -25 & -42 The product MN is: -23 & -28 \\ -25 & -42 ii) Find M^T + N^T. Step 1: Find the transpose of matrix M (M^T). The transpose of a matrix is obtained by interchanging its rows and columns. M^T = -3 & 5 \\ 4 & 6 Step 2: Find the transpose of matrix N (N^T). N^T = 1 & -5 \\ 0 & -7 Step 3: Add the transposed matrices M^T and N^T. To add matrices, we add their corresponding elements. M^T + N^T = -3 & 5 \\ 4 & 6 + 1 & -5 \\ 0 & -7 M^T + N^T = -3+1 & 5+(-5) \\ 4+0 & 6+(-7) M^T + N^T = -2 & 0 \\ 4 & -1 The sum M^T + N^T is: -2 & 0 \\ 4 & -1 iii) Find N^-1 (N inverse). Step 1: Calculate the determinant of matrix N. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. For N = 1 & 0 \\ -5 & -7 : (N) = (1)(-7) - (0)(-5) (N) = -7 - 0 (N) = -7 Step 2: Calculate the inverse of matrix N. For a 2 × 2 matrix a & b \\ c & d , the inverse is (1)/((A)) d & -b \\ -c & a . N^-1 = (1)/(-7) -7 & -0 \\ -(-5) & 1 N^-1 = -(1)/(7) -7 & 0 \\ 5 & 1 Step 3: Multiply each element by the scalar -(1)/(7). N^-1 = (-(1)/(7))(-7) & (-(1)/(7))(0) \\ (-(1)/(7))(5) & (-(1)/(7))(1) N^-1 = 1 & 0 \\ -(5)/(7) & -(1)/(7) The inverse of N is: 1 & 0 \\ -(5)/(7) & -(1)/(7)

Accepted Answer · 2026-04-01T19:51:56.943Z

Given the matrices: M = -3 & 4 \\ 5 & 6 N = 1 & 0 \\ -5 & -7 i) Find MN. Step 1: Multiply the matrices M and N. To multiply two matrices, we multiply the rows of the first matrix by the columns of the second matrix. MN = (-3)(1) + (4)(-5) & (-3)(0) + (4)(-7) \\ (5)(1) + (6)(-5) & (5)(0) + (6)(-7) Step 2: Perform the multiplications and additions. MN = -3 - 20 & 0 - 28 \\ 5 - 30 & 0 - 42 MN = -23 & -28 \\ -25 & -42 The product MN is: -23 & -28 \\ -25 & -42 ii) Find M^T + N^T. Step 1: Find the transpose of matrix M (M^T). The transpose of a matrix is obtained by interchanging its rows and columns. M^T = -3 & 5 \\ 4 & 6 Step 2: Find the transpose of matrix N (N^T). N^T = 1 & -5 \\ 0 & -7 Step 3: Add the transposed matrices M^T and N^T. To add matrices, we add their corresponding elements. M^T + N^T = -3 & 5 \\ 4 & 6 + 1 & -5 \\ 0 & -7 M^T + N^T = -3+1 & 5+(-5) \\ 4+0 & 6+(-7) M^T + N^T = -2 & 0 \\ 4 & -1 The sum M^T + N^T is: -2 & 0 \\ 4 & -1 iii) Find N^-1 (N inverse). Step 1: Calculate the determinant of matrix N. For a 2 × 2 matrix a & b \\ c & d , the determinant is ad - bc. For N = 1 & 0 \\ -5 & -7 : (N) = (1)(-7) - (0)(-5) (N) = -7 - 0 (N) = -7 Step 2: Calculate the inverse of matrix N. For a 2 × 2 matrix a & b \\ c & d , the inverse is (1)/((A)) d & -b \\ -c & a . N^-1 = (1)/(-7) -7 & -0 \\ -(-5) & 1 N^-1 = -(1)/(7) -7 & 0 \\ 5 & 1 Step 3: Multiply each element by the scalar -(1)/(7). N^-1 = (-(1)/(7))(-7) & (-(1)/(7))(0) \\ (-(1)/(7))(5) & (-(1)/(7))(1) N^-1 = 1 & 0 \\ -(5)/(7) & -(1)/(7) The inverse of N is: 1 & 0 \\ -(5)/(7) & -(1)/(7)

Given the matrices M = [[-3, 4], [5, 6]] and N = [[1, 0], [-5, -7]], find MN and M^T + N^T.

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Given the matrices M = [[-3, 4], [5, 6]] and N = [[1, 0], [-5, -7]], find MN and M^T + N^T.

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