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E=[[-1,4],[5,2]] and C=[[1,2],[-1,1]]. Let H=EC. Find H. H=[quad]

E=[−1452] \mathrm{E}=\left[\begin{array}{rr}-1 & 4 \\ 5 & 2\end{array}\right] and C=[12−11] C=\left[\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right] .\newlineLet H=EC \mathrm{H}=\mathrm{EC} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. E=[−1452] \mathrm{E}=\left[\begin{array}{rr}-1 & 4 \\ 5 & 2\end{array}\right] and C=[12−11] C=\left[\begin{array}{rr}1 & 2 \\ -1 & 1\end{array}\right] .\newlineLet H=EC \mathrm{H}=\mathrm{EC} . Find H \mathrm{H} .\newlineH= \mathbf{H}=
  1. Understand matrix multiplication: Understand matrix multiplication. To multiply two matrices, we take the dot product of the rows of the first matrix with the columns of the second matrix. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined.
  2. Set up matrix multiplication: Set up the multiplication of matrices EE and CC.E=[−14 52]E = \left[\begin{array}{cc}-1 & 4 \ 5 & 2\end{array}\right]C=[12 −11]C = \left[\begin{array}{cc}1 & 2 \ -1 & 1\end{array}\right]We will calculate the entries of the resulting matrix HH by multiplying the rows of EE by the columns of CC.
  3. Calculate H[1,1]H[1,1]: Calculate the entry H[1,1]H[1,1]. This is the dot product of the first row of EE with the first column of CC. H[1,1]=(−1×1)+(4×−1)=−1−4=−5H[1,1] = (-1 \times 1) + (4 \times -1) = -1 - 4 = -5
  4. Calculate H[1,2]H[1,2]: Calculate the entry H[1,2]H[1,2]. This is the dot product of the first row of EE with the second column of CC. H[1,2]=(−1×2)+(4×1)=−2+4=2H[1,2] = (-1 \times 2) + (4 \times 1) = -2 + 4 = 2
  5. Calculate H[2,1]H[2,1]: Calculate the entry H[2,1]H[2,1]. This is the dot product of the second row of EE with the first column of CC. H[2,1]=(5×1)+(2×−1)=5−2=3H[2,1] = (5 \times 1) + (2 \times -1) = 5 - 2 = 3
  6. Calculate H[2,2]H[2,2]: Calculate the entry H[2,2]H[2,2]. This is the dot product of the second row of EE with the second column of CC. H[2,2]=(5×2)+(2×1)=10+2=12H[2,2] = (5 \times 2) + (2 \times 1) = 10 + 2 = 12
  7. Combine results for matrix HH: Combine the results to form the matrix HH.H=[−52 312]H = \begin{bmatrix}-5 & 2 \ 3 & 12\end{bmatrix}

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