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B=[[0],[-1],[2]] and A=[[1,2]] Let H=BA. Find H. H=[]

B=[0−12] \mathrm{B}=\left[\begin{array}{r}0 \\ -1 \\ 2\end{array}\right] and A=[12] \mathrm{A}=\left[\begin{array}{ll}1 & 2\end{array}\right] \newlineLet H=BA \mathrm{H}=\mathrm{BA} . Find H \mathrm{H} .\newlineH= \mathbf{H}=

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Full solution

Q. B=[0−12] \mathrm{B}=\left[\begin{array}{r}0 \\ -1 \\ 2\end{array}\right] and A=[12] \mathrm{A}=\left[\begin{array}{ll}1 & 2\end{array}\right] \newlineLet H=BA \mathrm{H}=\mathrm{BA} . Find H \mathrm{H} .\newlineH= \mathbf{H}=
  1. Understand Matrix Dimensions: First, let's understand the dimensions of the matrices BB and AA. Matrix BB is a 3×13 \times 1 matrix and matrix AA is a 1×21 \times 2 matrix. To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. In this case, BB has 11 column and AA has 11 row, so we can multiply BB by AA.
  2. Perform Matrix Multiplication: Next, we perform the matrix multiplication BABA. The resulting matrix HH will have the same number of rows as BB and the same number of columns as AA, which means HH will be a 3×23 \times 2 matrix. We calculate the entries of HH by taking the dot product of the rows of BB with the columns of AA.
  3. Calculate Entries of Resulting Matrix: The calculation for each entry of HH is as follows:\newlineH[1,1]=B[1,1]×A[1,1]=0×1=0H[1,1] = B[1,1] \times A[1,1] = 0 \times 1 = 0\newlineH[1,2]=B[1,1]×A[1,2]=0×2=0H[1,2] = B[1,1] \times A[1,2] = 0 \times 2 = 0\newlineH[2,1]=B[2,1]×A[1,1]=−1×1=−1H[2,1] = B[2,1] \times A[1,1] = -1 \times 1 = -1\newlineH[2,2]=B[2,1]×A[1,2]=−1×2=−2H[2,2] = B[2,1] \times A[1,2] = -1 \times 2 = -2\newlineH[3,1]=B[3,1]×A[1,1]=2×1=2H[3,1] = B[3,1] \times A[1,1] = 2 \times 1 = 2\newlineH[3,2]=B[3,1]×A[1,2]=2×2=4H[3,2] = B[3,1] \times A[1,2] = 2 \times 2 = 4
  4. Write Down Resulting Matrix: Now we can write down the resulting matrix HH with the calculated entries: H=[00 −1−2 24]H = \begin{bmatrix} 0 & 0 \ -1 & -2 \ 2 & 4 \end{bmatrix}

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