Questions: Use matrices E and H to perform the operation: EH E=[[7 0], [3 -1], [-3 4]] H=[[-1 4], [6 2]] [[-7 28], [-9 10], [27 -4]]

Transcript text: Use matrices E and H to perform the operation: EH $E=\left[\begin{array}{cc}7 & 0 \\ 3 & -1 \\ -3 & 4\end{array}\right]$ $H=\left[\begin{array}{cc}-1 & 4 \\ 6 & 2\end{array}\right]$ $\left[\begin{array}{rr}-7 & 28 \\ -9 & 10 \\ 27 & -4\end{array}\right]$

Solution

Solution Steps

To perform the matrix multiplication $ EH $, we need to multiply matrix $ E $ by matrix $ H $. Matrix multiplication involves taking the dot product of rows from the first matrix with columns of the second matrix.

Step 1: Define Matrices $ E $ and $ H $

Given matrices: \[ E = \begin{bmatrix} 7 & 0 \\ 3 & -1 \\ -3 & 4 \end{bmatrix} \] \[ H = \begin{bmatrix} -1 & 4 \\ 6 & 2 \end{bmatrix} \]

Step 2: Perform Matrix Multiplication $ EH $

To find the product $ EH $, we multiply each element of the rows of $ E $ by the corresponding elements of the columns of $ H $ and sum the results.

Step 3: Calculate Each Element of the Resultant Matrix

The element at position $(i, j)$ in the resultant matrix is calculated as: \[ (EH)_{ij} = \sum_{k} E_{ik} \cdot H_{kj} \]

Calculation:

First row, first column: \[ (EH)_{11} = 7 \cdot (-1) + 0 \cdot 6 = -7 \]
First row, second column: \[ (EH)_{12} = 7 \cdot 4 + 0 \cdot 2 = 28 \]
Second row, first column: \[ (EH)_{21} = 3 \cdot (-1) + (-1) \cdot 6 = -3 - 6 = -9 \]
Second row, second column: \[ (EH)_{22} = 3 \cdot 4 + (-1) \cdot 2 = 12 - 2 = 10 \]
Third row, first column: \[ (EH)_{31} = (-3) \cdot (-1) + 4 \cdot 6 = 3 + 24 = 27 \]
Third row, second column: \[ (EH)_{32} = (-3) \cdot 4 + 4 \cdot 2 = -12 + 8 = -4 \]