Example Question - matrix determinant
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Matrix Determinant Calculation
<p>The provided photo seems to show the determinant of a 3x3 matrix being calculated. However, the step-by-step calculation is not fully visible, and there's some text in another language at the bottom. With the visible content, I will assume the goal is to calculate the determinant of a matrix displayed at the top right corner of the image.</p> <p>Let the matrix A be:</p> \[ A = \begin{bmatrix} -3 & 1 & 5 \\ 1 & -3 & -5 \\ 1 & 3 & 5 \\ \end{bmatrix} \] <p>To find the determinant of matrix A (denoted as det(A) or |A|), we can use the following formula for a 3x3 matrix:</p> <p>det(A) = a_{11}(a_{22}a_{33} - a_{23}a_{32}) - a_{12}(a_{21}a_{33} - a_{23}a_{31}) + a_{13}(a_{21}a_{32} - a_{22}a_{31})</p> <p>Substituting the values from matrix A, we get:</p> \[ \begin{align*} \text{det}(A) &= (-3)((-3)(5) - (-5)(3)) - (1)((1)(5) - (3)(-5)) + (5)((1)(3) - (-3)(1)) \\ &= (-3)((-15) - (-15)) - (1)((5) - (-15)) + (5)((3) - (-3)) \\ &= (-3)(0) - (1)(20) + (5)(6) \\ &= 0 - 20 + 30 \\ &= 10 \end{align*} \] <p>Therefore, the determinant of matrix A is 10.</p>
Matrix Determinant Calculation Using the Laplace Expansion
<p>Determinant calculation of a 3x3 matrix using the Laplace expansion.</p> <p>Given Matrix A:</p> <p>\[ A = \begin{bmatrix} -1 & 3 & 5 \\ 1 & -3 & -5 \\ 1 & 3 & 5 \end{bmatrix} \]</p> <p>Expanding along the first row:</p> <p>\[ \text{det}(A) = (-1)^{1+1} \cdot (-1) \cdot \begin{vmatrix} -3 & -5 \\ 3 & 5 \end{vmatrix} + (-1)^{1+2} \cdot 3 \cdot \begin{vmatrix} 1 & -5 \\ 1 & 5 \end{vmatrix} + (-1)^{1+3} \cdot 5 \cdot \begin{vmatrix} 1 & -3 \\ 1 & 3 \end{vmatrix} \]</p> <p>\[ \text{det}(A) = -1 \cdot ((-3)\cdot(5) - (-5)\cdot(3)) + 3 \cdot (1\cdot(5) - (-5)\cdot(1)) + (-5) \cdot (1\cdot(3) - (-3)\cdot(1)) \]</p> <p>\[ \text{det}(A) = -1 \cdot (-15 + 15) + 3 \cdot (5 + 5) - 5 \cdot (3 + 3) \]</p> <p>\[ \text{det}(A) = -1 \cdot 0 + 3 \cdot 10 - 5 \cdot 6 \]</p> <p>\[ \text{det}(A) = 0 + 30 - 30 \]</p> <p>\[ \text{det}(A) = 0 \]</p> <p>Hence, the determinant of matrix A is 0.</p>
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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