Example Question - 2x2 matrix
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Matrix Multiplication: Product of Matrices A and B
The question is asking us to calculate the product of two matrices A and B: A = \(\begin{pmatrix} 3 & 1 \\ 5 & 4 \end{pmatrix}\) B = \(\begin{pmatrix} 5 & 2 & -1\\ 1 & 3 & 4 \end{pmatrix}\) Matrix A is a 2x2 matrix and matrix B is a 2x3 matrix. We can multiply A and B because the number of columns in A (which is 2) matches the number of rows in B (which is also 2). The resulting matrix will have the same number of rows as matrix A and the same number of columns as matrix B, which means the product will be a 2x3 matrix. Here's how we calculate the product AB: AB = \(\begin{pmatrix} 3 & 1 \\ 5 & 4 \end{pmatrix} \times \begin{pmatrix} 5 & 2 & -1\\ 1 & 3 & 4 \end{pmatrix}\) AB = \(\begin{pmatrix} 3*5 + 1*1 & 3*2 + 1*3 & 3*(-1) + 1*4 \\ 5*5 + 4*1 & 5*2 + 4*3 & 5*(-1) + 4*4 \end{pmatrix}\) AB = \(\begin{pmatrix} 15 + 1 & 6 + 3 & -3 + 4 \\ 25 + 4 & 10 + 12 & -5 + 16 \end{pmatrix}\) AB = \(\begin{pmatrix} 16 & 9 & 1 \\ 29 & 22 & 11 \end{pmatrix}\) So the product of matrices A and B is the 2x3 matrix: \(\begin{pmatrix} 16 & 9 & 1 \\ 29 & 22 & 11 \end{pmatrix}\)
Determinant calculation for 2x2 matrix
The given matrix M is: \[ M = \begin{pmatrix} 2 & x \\ -1 & 3 \end{pmatrix} \] And we are told that the determinant of matrix M is 9. The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is calculated by \(ad - bc\). Therefore, for the given matrix M, the determinant is calculated by: \( (2)(3) - (-1)(x) = 6 + x \) Since we know that the determinant is 9, we can set up the following equation: \( 6 + x = 9 \) Now, let's solve for \( x \): \( x = 9 - 6 \) \( x = 3 \) Hence, the value of \( x \) that satisfies the given determinant value is 3.
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