Example Question - 3x3 matrix
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Matrix Multiplication Calculation Result
The image shows the handwritten task of calculating the matrix product A * I, where matrix A is a 3x3 matrix with entries: \[ A = \begin{bmatrix} 3 & -5 & 6 \\ 2 & 4 & 2 \\ 7 & 0 & 3 \end{bmatrix} \] and I is the 3x3 identity matrix, which is: \[ I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} \] Multiplying any matrix by the identity matrix yields the original matrix. So the product A * I is just the matrix A itself: \[ A * I = A = \begin{bmatrix} 3 & -5 & 6 \\ 2 & 4 & 2 \\ 7 & 0 & 3 \end{bmatrix} \]
Understanding Square Matrices
A square matrix is a matrix that has the same number of rows and columns. We can determine if a matrix is square by checking their dimensions. Let's look at the matrices one by one: Matrix A: \[\begin{bmatrix} -1 & 4 & -2 \\ -4 & 3 & 5 \\ -3 & 1 & 6 \end{bmatrix}\] Matrix A has 3 rows and 3 columns, so it is a 3x3 matrix. Matrix B: \[\begin{bmatrix} 5 & 2 & -1 \\ 1 & 3 & 4 \\ \end{bmatrix}\] Matrix B has 2 rows and 3 columns, so it is a 2x3 matrix and not square. Matrix C: \[\begin{bmatrix} 12 & 6 \\ 5 & 3 \\ 0 & 7 \\ \end{bmatrix}\] Matrix C has 3 rows and 2 columns, so it is a 3x2 matrix and not square. Only Matrix A is a square matrix because it has the same number of rows and columns (3x3).
Determinant Calculation for 3x3 Matrix
The image contains a determinant of a 3x3 matrix equal to -30. We need to calculate the value(s) of x for which this determinant holds true. The determinant is: | 3 1 x | | -1 3 4 | | x 1 0 | To find the determinant, we will expand along the third column (as it seems to be the easiest due to the zero present): det = x * | -1 3 | | x 1 | - x * | 3 1 | | x 0 | + 4 * | 3 1 | | -1 3 | Solving each of these determinants: det(x) = x((-1)(1) - (3)(x)) - x((3)(0) - (1)(x)) + 4((3)(3) - (1)(-1)) det(x) = x(-1 - 3x) - x(0 - x) + 4(9 + 1) det(x) = x(-1 - 3x) - x(-x) + 4(10) det(x) = -x - 3x^2 + x^2 + 40 det(x) = -3x^2 + x^2 - x + 40 Combine like terms: det(x) = -2x^2 - x + 40 Now we set this determinant equal to the given value (-30) and solve for x: -2x^2 - x + 40 = -30 Move -30 to the other side: -2x^2 - x + 40 + 30 = 0 -2x^2 - x + 70 = 0 This is a quadratic equation, which can often be solved by factoring, completing the square, or using the quadratic formula. To use the quadratic formula (\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)) we identify the coefficients a = -2, b = -1, c = 70. Before using the quadratic formula, it might be helpful to check if the quadratic can be factored easily, but in this case, it cannot be factored into integers, so let's apply the quadratic formula: \( x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(-2)(70)}}{2(-2)} \) \( x = \frac{1 \pm \sqrt{1 + 560}}{-4} \) \( x = \frac{1 \pm \sqrt{561}}{-4} \) The two solutions for x are: \( x = \frac{1 + \sqrt{561}}{-4} \) and \( x = \frac{1 - \sqrt{561}}{-4} \) These are the values of x for which the determinant of the matrix equals -30.
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