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E=[[1,-1,1],[2,3,4]] and A=[[2,4],[2,-2],[-1,5]] Let H=EA. Find H. H=[◻]◻

E=[1−11234] \mathrm{E}=\left[\begin{array}{rrr}1 & -1 & 1 \\ 2 & 3 & 4\end{array}\right] and A=[242−2−15] A=\left[\begin{array}{rr}2 & 4 \\ 2 & -2 \\ -1 & 5\end{array}\right] \newlineLet H=EA \mathrm{H}=\mathrm{EA} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. E=[1−11234] \mathrm{E}=\left[\begin{array}{rrr}1 & -1 & 1 \\ 2 & 3 & 4\end{array}\right] and A=[242−2−15] A=\left[\begin{array}{rr}2 & 4 \\ 2 & -2 \\ -1 & 5\end{array}\right] \newlineLet H=EA \mathrm{H}=\mathrm{EA} . Find H \mathrm{H} .\newlineH= \mathbf{H}=
  1. Understand matrix multiplication: Understand matrix multiplication.\newlineTo multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix.
  2. Verify matrix dimensions: Verify the dimensions of matrices EE and AA. Matrix EE has dimensions 2×32 \times 3 (22 rows and 33 columns), and matrix AA has dimensions 3×23 \times 2 (33 rows and 22 columns). Since the number of columns in EE is equal to the number of rows in AA, we can multiply these matrices.
  3. Set up multiplication: Set up the multiplication of matrices EE and AA. To multiply EE by AA, we will calculate the sum of the products of the corresponding entries from the rows of EE and the columns of AA.
  4. Calculate first entry of H: Calculate the first entry of matrix HH. For the entry in the first row and first column of HH, we multiply the corresponding elements of the first row of EE with the first column of AA and sum them up: H[1,1]=(1×2)+(−1×2)+(1×−1)=2−2−1=−1H[1,1] = (1 \times 2) + (-1 \times 2) + (1 \times -1) = 2 - 2 - 1 = -1
  5. Calculate second entry of H: Calculate the second entry of the first row of matrix HH. For the entry in the first row and second column of HH, we multiply the corresponding elements of the first row of EE with the second column of AA and sum them up: H[1,2]=(1×4)+(−1×−2)+(1×5)=4+2+5=11H[1,2] = (1\times4) + (-1\times-2) + (1\times5) = 4 + 2 + 5 = 11
  6. Calculate first entry of second row: Calculate the first entry of the second row of matrix HH. For the entry in the second row and first column of HH, we multiply the corresponding elements of the second row of EE with the first column of AA and sum them up: H[2,1]=(2×2)+(3×2)+(4×−1)=4+6−4=6H[2,1] = (2 \times 2) + (3 \times 2) + (4 \times -1) = 4 + 6 - 4 = 6
  7. Calculate second entry of second row: Calculate the second entry of the second row of matrix HH. For the entry in the second row and second column of HH, we multiply the corresponding elements of the second row of EE with the second column of AA and sum them up: H[2,2]=(2×4)+(3×−2)+(4×5)=8−6+20=22H[2,2] = (2 \times 4) + (3 \times -2) + (4 \times 5) = 8 - 6 + 20 = 22
  8. Combine results for matrix HH: Combine the results to form matrix HH. We combine the results from steps 44 to 77 to form the resulting matrix HH: H=[[−1,11],[6,22]]H = [[-1, 11], [6, 22]]

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