$\mathrm{E}=\left[\begin{array}{lll}0 & 3 & 5 \\ 5 & 5 & 2\end{array}\right]$ and $\mathrm{D}=\left[\begin{array}{rr}3 & 4 \\ 3 & -2 \\ 4 & -2\end{array}\right]$ $\newline$ Let $\mathrm{H}=\mathrm{ED}$ . Find $\mathrm{H}$ . $\newline$ $\mathbf{H}=$

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Precalculus

Find trigonometric ratios using a Pythagorean or reciprocal identity

Full solution

\mathrm{E}=\left[\begin{array}{lll}0 & 3 & 5 \\ 5 & 5 & 2\end{array}\right]

and

\mathrm{D}=\left[\begin{array}{rr}3 & 4 \\ 3 & -2 \\ 4 & -2\end{array}\right]

\newline

Let

\mathrm{H}=\mathrm{ED}

. Find

\mathrm{H}

\newline

\mathbf{H}=

Understand matrix multiplication: Understand matrix multiplication. $\newline$ To multiply two matrices, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Matrix $E$ is a $2 \times 3$ matrix and matrix $D$ is a $3 \times 2$ matrix, so they can be multiplied to result in a $2 \times 2$ matrix $H$ .
Set up the multiplication: Set up the multiplication. $\newline$ We will calculate each element of matrix $H$ by taking the dot product of the corresponding row from matrix $E$ and the corresponding column from matrix $D$ .
Calculate first element of H: Calculate the first element of matrix H $H[1,1]$ .
$H[1,1] = E[1,1] \times D[1,1] + E[1,2] \times D[2,1] + E[1,3] \times D[3,1]$
$H[1,1] = 0 \times 3 + 3 \times 3 + 5 \times 4$
$H[1,1] = 0 + 9 + 20$
$H[1,1] = 29$
Calculate second element of H: Calculate the second element of matrix H $H[1,2]$ . $H[1,2] = E[1,1]\cdot D[1,2] + E[1,2]\cdot D[2,2] + E[1,3]\cdot D[3,2]$ $H[1,2] = 0\cdot 4 + 3\cdot (-2) + 5\cdot (-2)$ $H[1,2] = 0 - 6 - 10$ $H[1,2] = -16$
Calculate third element of H: Calculate the third element of matrix H $H[2,1]$ . $H[2,1] = E[2,1]\cdot D[1,1] + E[2,2]\cdot D[2,1] + E[2,3]\cdot D[3,1]$ $H[2,1] = 5\cdot 3 + 5\cdot 3 + 2\cdot 4$ $H[2,1] = 15 + 15 + 8$ $H[2,1] = 38$
Calculate fourth element of H: Calculate the fourth element of matrix $H$ ( $H[2,2]$ ). $\newline$ $H[2,2] = E[2,1]\cdot D[1,2] + E[2,2]\cdot D[2,2] + E[2,3]\cdot D[3,2]$ $\newline$ $H[2,2] = 5\cdot 4 + 5\cdot (-2) + 2\cdot (-2)$ $\newline$ $H[2,2] = 20 - 10 - 4$ $\newline$ $H[2,2] = 6$
Combine elements for matrix H: Combine the elements to form matrix H. $\newline$ H = \left[\begin{array}{cc}$\newlineH[1,1] & H[1,2] (\newline$H[2,1] & H[2,2] $\newline$ \end{array}\right]\) $\newline$ H = \left[\begin{array}{cc}$\newline29 & -16 (\newline$38 & 6 $\newline$ \end{array}\right]\)