6. Suppose that \( A \) is a \( 2 \times 2 \) matrix, \( B \) is a \( 3 \times 2 \) matrix and \( C \) is a \( 2 \times 1 \) matrix. Which one of the following statements is correct? The matrix \( A B C \) is \( 2 \times 1 \) The matrix \( B A C \) is \( 3 \times 1 \) The matrix \( B C A \) is \( 3 \times 2 \) The matrix \( A B C \) is \( 3 \times 1 \) The matrix \( B A C \) is \( 2 \times 1 \)
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- \( B \) is a \( 3 \times 2 \) matrix. - \( C \) is a \( 2 \times 1 \) matrix. Show more…
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Problem 4. If $A=\left(\begin{array}{rr}-3 & 0 \\ 7 & -4\end{array}\right)$ $B=\left(\begin{array}{rr}2 & -1 \\ -7 & 4\end{array}\right)$ and $C=\left(\begin{array}{rr}1 & 0 \\ -2 & -4\end{array}\right)$ find $2 A-3 B+4 C$ For scalar multiplication, each element is multiplied by the scalar quantity, hence $$ \begin{aligned} &2 A=2\left(\begin{array}{rr} -3 & 0 \\ 7 & -4 \end{array}\right)=\left(\begin{array}{rr} -6 & 0 \\ 14 & -8 \end{array}\right) \\ &3 B=3\left(\begin{array}{rr} 2 & -1 \\ -7 & 4 \end{array}\right)=\left(\begin{array}{rr} 6 & -3 \\ -21 & 12 \end{array}\right) \end{aligned} $$ and $4 C=4\left(\begin{array}{rr}1 & 0 \\ -2 & -4\end{array}\right)=\left(\begin{array}{rr}4 & 0 \\ -8 & -16\end{array}\right)$ Hence $2 A-3 B+4 C$ $$ \begin{aligned} &=\left(\begin{array}{rr} -6 & 0 \\ 14 & -8 \end{array}\right)-\left(\begin{array}{rr} 6 & -3 \\ -21 & 12 \end{array}\right)+\left(\begin{array}{rr} 4 & 0 \\ -8 & -16 \end{array}\right) \\ &=\left(\begin{array}{cc} -6-6+4 & 0-(-3)+0 \\ 14-(-21)+(-8) & -8-12+(-16) \end{array}\right) \\ &=\left(\begin{array}{rr} -8 & 3 \\ 27 & -36 \end{array}\right) \end{aligned} $$ When a matrix $A$ is multiplied by another matrix $B$, a single matrix results in which elements are obtained from the sum of the products of the corresponding rows of $A$ and the corresponding columns of $B$. Two matrices $A$ and $B$ may be multiplied together, provided the number of elements in the rows of matrix $A$ are equal to the number of elements in the columns of matrix $B$, In general terms, when multiplying a matrix of dimensions $(m$ by $n$ ) by a matrix of dimensions ( $n$ by $r$ ), the resulting matrix has dimensions ( $m$ by $r$ ). Thus a 2 by 3 matrix multiplied by a 3 by 1 matrix gives a matrix of dimensions 2 by 1 .
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Let the matrices A and C be given as follows: Which one of the following is wrong? A+B is not possible AB has the same size (dimension) as matrix A BAT has the same size (dimension) as C C.A is 2x2
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