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D=[[1,0],[-1,5]] and B=[[1,0,2],[4,5,4]] Let H=DB. Find H. H=[quad]

D=[10−15] \mathrm{D}=\left[\begin{array}{rr}1 & 0 \\ -1 & 5\end{array}\right] and B=[102454] \mathrm{B}=\left[\begin{array}{lll} 1 & 0 & 2 \\ 4 & 5 & 4 \end{array}\right] \newlineLet H=DB \mathrm{H}=\mathrm{DB} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. D=[10−15] \mathrm{D}=\left[\begin{array}{rr}1 & 0 \\ -1 & 5\end{array}\right] and B=[102454] \mathrm{B}=\left[\begin{array}{lll} 1 & 0 & 2 \\ 4 & 5 & 4 \end{array}\right] \newlineLet H=DB \mathrm{H}=\mathrm{DB} . 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 BB and performing dot products to get the entries of the resulting 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, DD is a 2×22 \times 2 matrix and BB is a 2×32 \times 3 matrix, so the multiplication is defined and will result in a 2×32 \times 3 matrix.
  2. Perform first row multiplication: Perform the multiplication for the first row of HH. The first row of HH will be obtained by multiplying the first row of DD by each column of BB. H[1,1]=(1×1)+(0×4)=1H[1,1] = (1 \times 1) + (0 \times 4) = 1 H[1,2]=(1×0)+(0×5)=0H[1,2] = (1 \times 0) + (0 \times 5) = 0 H[1,3]=(1×2)+(0×4)=2H[1,3] = (1 \times 2) + (0 \times 4) = 2 So the first row of HH is [1,0,2][1, 0, 2].
  3. Perform second row multiplication: Perform the multiplication for the second row of HH. The second row of HH will be obtained by multiplying the second row of DD by each column of BB. H[2,1]=(−1×1)+(5×4)=−1+20=19H[2,1] = (-1 \times 1) + (5 \times 4) = -1 + 20 = 19 H[2,2]=(−1×0)+(5×5)=0+25=25H[2,2] = (-1 \times 0) + (5 \times 5) = 0 + 25 = 25 H[2,3]=(−1×2)+(5×4)=−2+20=18H[2,3] = (-1 \times 2) + (5 \times 4) = -2 + 20 = 18 So the second row of HH is [19,25,18][19, 25, 18].

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