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E=[[5,5,1],[4,0,3]] and B=[[-1,5],[1,2],[-2,3]] Let H=EB. Find H. H=

E=[551403] \mathrm{E}=\left[\begin{array}{lll}5 & 5 & 1 \\ 4 & 0 & 3\end{array}\right] and B=[−1512−23] \mathrm{B}=\left[\begin{array}{rr}-1 & 5 \\ 1 & 2 \\ -2 & 3\end{array}\right] \newlineLet H=EB \mathrm{H}=\mathrm{EB} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. E=[551403] \mathrm{E}=\left[\begin{array}{lll}5 & 5 & 1 \\ 4 & 0 & 3\end{array}\right] and B=[−1512−23] \mathrm{B}=\left[\begin{array}{rr}-1 & 5 \\ 1 & 2 \\ -2 & 3\end{array}\right] \newlineLet H=EB \mathrm{H}=\mathrm{EB} . 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.\newlineEE is a 2×32 \times 3 matrix and BB is a 3×23 \times 2 matrix, so we can multiply EE by BB to get a 2×22 \times 2 matrix HH.
  2. Set up the multiplication: Set up the multiplication.\newlineWe will calculate each element of matrix HH by taking the dot product of the corresponding row from EE and the corresponding column from BB.
  3. Calculate first element of H: Calculate the first element of matrix HH.H[1,1]=(5×−1)+(5×1)+(1×−2)=−5+5−2=−2H[1,1] = (5 \times -1) + (5 \times 1) + (1 \times -2) = -5 + 5 - 2 = -2
  4. Calculate second element of first row: Calculate the second element of the first row of matrix HH.H[1,2]=(5×5)+(5×2)+(1×3)=25+10+3=38H[1,2] = (5 \times 5) + (5 \times 2) + (1 \times 3) = 25 + 10 + 3 = 38
  5. Calculate first element of second row: Calculate the first element of the second row of matrix HH.H[2,1]=(4×−1)+(0×1)+(3×−2)=−4+0−6=−10H[2,1] = (4 \times -1) + (0 \times 1) + (3 \times -2) = -4 + 0 - 6 = -10
  6. Calculate second element of second row: Calculate the second element of the second row of matrix HH.H[2,2]=(4×5)+(0×2)+(3×3)=20+0+9=29H[2,2] = (4 \times 5) + (0 \times 2) + (3 \times 3) = 20 + 0 + 9 = 29
  7. Combine results for matrix HH: Combine the results to form matrix HH.H = \begin{bmatrix}-2 & 38\-10 & 29\end{bmatrix}

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