Example Question - matrix addition
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Matrix Addition and Equations
<p>First, add the corresponding elements of the matrices:</p> <p>\(\begin{bmatrix} 0 & 1 \\ y & 5 \end{bmatrix} + \begin{bmatrix} 4 & -1 \\ 6 & x \end{bmatrix} = \begin{bmatrix} 0 + 4 & 1 - 1 \\ y + 6 & 5 + x \end{bmatrix}\)</p> <p>Which simplifies to:</p> <p>\(\begin{bmatrix} 4 & 0 \\ y + 6 & 5 + x \end{bmatrix}\)</p> <p>Set the resulting matrix equal to the given matrix:</p> <p>\(\begin{bmatrix} 4 & 0 \\ x & 7 \end{bmatrix}\)</p> <p>This gives the equations:</p> <p>1. \(y + 6 = x\)</p> <p>2. \(5 + x = 7\)</p> <p>From the second equation, solve for \(x\):</p> <p>\(x = 7 - 5 = 2\)</p> <p>Now substitute \(x = 2\) into the first equation:</p> <p>\(y + 6 = 2 \implies y = 2 - 6 = -4\)</p> <p>Thus, the solution is:</p> <p>\(y = -4\), \(x = 2\)</p>
Matrix Multiplication and Addition Exercise
To solve this problem, we need to multiply matrix A by 2 and then add the resulting matrix to matrix B. We'll multiply each entry of matrix A by 2 and then add the corresponding entries of matrix B to these results. Matrix A: \[ \begin{pmatrix} 5 & 1 \\ 2 & 3 \\ 1 & 4 \end{pmatrix} \] Matrix B: \[ \begin{pmatrix} 2 & 4 \\ 1 & -3 \\ 2 & -1 \end{pmatrix} \] Step 1: Multiply each entry of matrix A by 2: \[ 2 \times A = 2 \times \begin{pmatrix} 5 & 1 \\ 2 & 3 \\ 1 & 4 \end{pmatrix} = \begin{pmatrix} 2 \times 5 & 2 \times 1 \\ 2 \times 2 & 2 \times 3 \\ 2 \times 1 & 2 \times 4 \end{pmatrix} = \begin{pmatrix} 10 & 2 \\ 4 & 6 \\ 2 & 8 \end{pmatrix} \] Step 2: Add matrices 2A and B: \[ 2A + B = \begin{pmatrix} 10 & 2 \\ 4 & 6 \\ 2 & 8 \end{pmatrix} + \begin{pmatrix} 2 & 4 \\ 1 & -3 \\ 2 & -1 \end{pmatrix} = \begin{pmatrix} 10+2 & 2+4 \\ 4+1 & 6-3 \\ 2+2 & 8-1 \end{pmatrix} = \begin{pmatrix} 12 & 6 \\ 5 & 3 \\ 4 & 7 \end{pmatrix} \] The resulting matrix is: \[ \begin{pmatrix} 12 & 6 \\ 5 & 3 \\ 4 & 7 \end{pmatrix} \] By comparing this result with the options provided, we can see that the answer is option C.
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