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F=[[0,4],[5,2]]" and "D=[[2,5],[3,0]] Let H=FD. Find H. H=[]

F=[0452] and D=[2530] \mathrm{F}=\left[\begin{array}{ll} 0 & 4 \\ 5 & 2 \end{array}\right] \text { and } \mathrm{D}=\left[\begin{array}{ll} 2 & 5 \\ 3 & 0 \end{array}\right] \newlineLet H=FD \mathrm{H}=\mathrm{FD} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. F=[0452] and D=[2530] \mathrm{F}=\left[\begin{array}{ll} 0 & 4 \\ 5 & 2 \end{array}\right] \text { and } \mathrm{D}=\left[\begin{array}{ll} 2 & 5 \\ 3 & 0 \end{array}\right] \newlineLet H=FD \mathrm{H}=\mathrm{FD} . 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 be equal to the number of rows in the second matrix for multiplication to be possible.
  2. Set up matrix multiplication: Set up the multiplication of matrix FF by matrix DD. Matrix FF is a 2×22 \times 2 matrix and matrix DD is also a 2×22 \times 2 matrix, so their multiplication is possible and will result in another 2×22 \times 2 matrix HH.
  3. Calculate first element of H: Calculate the first element of matrix HH. The first element of matrix HH (H[1,1]H[1,1]) is the dot product of the first row of FF and the first column of DD. H[1,1]=F[1,1]×D[1,1]+F[1,2]×D[2,1]=0×2+4×3=0+12=12H[1,1] = F[1,1]\times D[1,1] + F[1,2]\times D[2,1] = 0\times 2 + 4\times 3 = 0 + 12 = 12
  4. Calculate second element of H: Calculate the second element of matrix H. The second element of matrix H H[1,2]H[1,2] is the dot product of the first row of F and the second column of D. H[1,2]=F[1,1]×D[1,2]+F[1,2]×D[2,2]=0×5+4×0=0+0=0H[1,2] = F[1,1]\times D[1,2] + F[1,2]\times D[2,2] = 0\times 5 + 4\times 0 = 0 + 0 = 0
  5. Calculate third element of H: Calculate the third element of matrix HH. The third element of matrix HH (H[2,1]H[2,1]) is the dot product of the second row of FF and the first column of DD. H[2,1]=F[2,1]⋅D[1,1]+F[2,2]⋅D[2,1]=5⋅2+2⋅3=10+6=16H[2,1] = F[2,1]\cdot D[1,1] + F[2,2]\cdot D[2,1] = 5\cdot 2 + 2\cdot 3 = 10 + 6 = 16
  6. Calculate fourth element of H: Calculate the fourth element of matrix H.\newlineThe fourth element of matrix H H[2,2]H[2,2] is the dot product of the second row of F and the second column of D.\newlineH[2,2]=F[2,1]⋅D[1,2]+F[2,2]⋅D[2,2]=5⋅5+2⋅0=25+0=25H[2,2] = F[2,1]\cdot D[1,2] + F[2,2]\cdot D[2,2] = 5\cdot 5 + 2\cdot 0 = 25 + 0 = 25

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