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D=[[-1,0,4],[2,0,0]] and F=[[-1,1],[-1,3],[2,4]] Let H=DF. Find H. H=[]

D=[−104200] \mathrm{D}=\left[\begin{array}{rrr}-1 & 0 & 4 \\ 2 & 0 & 0\end{array}\right] and F=[−11−1324] F=\left[\begin{array}{rr}-1 & 1 \\ -1 & 3 \\ 2 & 4\end{array}\right] \newlineLet H=DF \mathrm{H}=\mathrm{DF} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. D=[−104200] \mathrm{D}=\left[\begin{array}{rrr}-1 & 0 & 4 \\ 2 & 0 & 0\end{array}\right] and F=[−11−1324] F=\left[\begin{array}{rr}-1 & 1 \\ -1 & 3 \\ 2 & 4\end{array}\right] \newlineLet H=DF \mathrm{H}=\mathrm{DF} . Find H \mathrm{H} .\newlineH= \mathbf{H}=
  1. Understand matrix multiplication: Understand matrix multiplication. Matrix multiplication involves taking the rows of the first matrix DD and columns of the second matrix FF and performing dot products to get the entries of the new matrix HH. The number of columns in the first matrix must equal the number of rows in the second matrix for the multiplication to be defined. In this case, matrix DD is a 2×32 \times 3 matrix and matrix FF is a 3×23 \times 2 matrix, so their product will be a 2×22 \times 2 matrix.
  2. Calculate first entry of H: Calculate the first entry of matrix HH. The first entry of HH is the dot product of the first row of DD and the first column of FF: H[1,1]=D[1,1]⋅F[1,1]+D[1,2]⋅F[2,1]+D[1,3]⋅F[3,1]H[1,1] = D[1,1]\cdot F[1,1] + D[1,2]\cdot F[2,1] + D[1,3]\cdot F[3,1] H[1,1]=(−1)⋅(−1)+(0)⋅(−1)+(4)⋅(2)H[1,1] = (-1)\cdot(-1) + (0)\cdot(-1) + (4)\cdot(2) H[1,1]=1+0+8H[1,1] = 1 + 0 + 8 H[1,1]=9H[1,1] = 9
  3. Calculate second entry of H: Calculate the second entry of matrix H.\newlineThe second entry of H is the dot product of the first row of D and the second column of F:\newlineH[1,2]=D[1,1]⋅F[1,2]+D[1,2]⋅F[2,2]+D[1,3]⋅F[3,2]H[1,2] = D[1,1]\cdot F[1,2] + D[1,2]\cdot F[2,2] + D[1,3]\cdot F[3,2]\newlineH[1,2]=(−1)⋅(1)+(0)⋅(3)+(4)⋅(4)H[1,2] = (-1)\cdot(1) + (0)\cdot(3) + (4)\cdot(4)\newlineH[1,2]=−1+0+16H[1,2] = -1 + 0 + 16\newlineH[1,2]=15H[1,2] = 15
  4. Calculate third entry of H: Calculate the third entry of matrix H.\newlineThe third entry of H is the dot product of the second row of D and the first column of F:\newlineH[2,1]=D[2,1]⋅F[1,1]+D[2,2]⋅F[2,1]+D[2,3]⋅F[3,1]H[2,1] = D[2,1]\cdot F[1,1] + D[2,2]\cdot F[2,1] + D[2,3]\cdot F[3,1]\newlineH[2,1]=(2)⋅(−1)+(0)⋅(−1)+(0)⋅(2)H[2,1] = (2)\cdot(-1) + (0)\cdot(-1) + (0)\cdot(2)\newlineH[2,1]=−2+0+0H[2,1] = -2 + 0 + 0\newlineH[2,1]=−2H[2,1] = -2
  5. Calculate fourth entry of H: Calculate the fourth entry of matrix H.\newlineThe fourth entry of H is the dot product of the second row of D and the second column of F:\newlineH[2,2]=D[2,1]⋅F[1,2]+D[2,2]⋅F[2,2]+D[2,3]⋅F[3,2]H[2,2] = D[2,1]\cdot F[1,2] + D[2,2]\cdot F[2,2] + D[2,3]\cdot F[3,2]\newlineH[2,2]=(2)⋅(1)+(0)⋅(3)+(0)⋅(4)H[2,2] = (2)\cdot(1) + (0)\cdot(3) + (0)\cdot(4)\newlineH[2,2]=2+0+0H[2,2] = 2 + 0 + 0\newlineH[2,2]=2H[2,2] = 2
  6. Combine results to form HH: Combine the results to form matrix HH. We have calculated all four entries of the matrix HH, so we can now form the matrix: H=[915 −22]H = \left[\begin{array}{cc} 9 & 15 \ -2 & 2 \end{array}\right]

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