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calculer le produit de $AB^T$ quand $A = \begin{pmatrix} -1 & 0 \ 3 & 2 \end{pmatrix} ;B = \begin{pmatrix} 4 & -3 \ 0 & 2 \ 1 & 0 \end{pmatrix}$

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Composition of linear and quadratic functions: find a value

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Q. calculer le produit de

AB^T

quand

A = \begin{pmatrix} -1 & 0 \ 3 & 2 \end{pmatrix} ;B = \begin{pmatrix} 4 & -3 \ 0 & 2 \ 1 & 0 \end{pmatrix}

Define matrices $A$ and $B$ : Define matrices $A$ and $B$ . $A = \left[ \begin{array}{cc} -1 & 0 \ 3 & 2 \end{array} \right]$ $B = \left[ \begin{array}{cc} 4 & -3 \ 0 & 2 \ 1 & 0 \end{array} \right]$
Calculate transpose of $B$ : Calculate the transpose of matrix $B$ ( $B^T$ ). $\newline$ $B^T = \left[ \begin{array}{ccc} 4 & 0 & 1 \ -3 & 2 & 0 \end{array} \right]$
Multiply $A$ by $B^T$ : Multiply matrix $A$ by $B^T$ . $\newline$ To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. $A$ is $2 \times 2$ and $B^T$ is $2 \times 3$ , so we can multiply them. $\newline$ Perform the multiplication: $\newline$ $1$ st row of $A$ * $1$ st column of $B^T$ = $B^T$ $0$ = $B^T$ $1$ $\newline$ $1$ st row of $A$ * $2$ nd column of $B^T$ = $B^T$ $4$ = $B^T$ $5$ $\newline$ $1$ st row of $A$ * $3$ rd column of $B^T$ = $B^T$ $8$ = $B^T$ $9$ $\newline$ $2$ nd row of $A$ * $1$ st column of $B^T$ = $A$ $2$ = $A$ $3$ $\newline$ $2$ nd row of $A$ * $2$ nd column of $B^T$ = $A$ $6$ = $A$ $7$ $\newline$ $2$ nd row of $A$ * $3$ rd column of $B^T$ = $B^T$ $0$ = $B^T$ $1$ $\newline$ Resulting matrix $B^T$ $2$

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