Example Question - determinant
Here are examples of questions we've helped users solve.
Determinants of Matrices with Variable Elements
Since this image contains determinant calculations and operations on matrices with variables, I will provide the solution to find the determinant based on the visible matrices in the image. <p>\text{Let's calculate the determinant of the first visible matrix:}</p> <p>\begin{vmatrix} a-1 & 3 & 5 \\ -1 & 3-5 & 5 \\ 1 & 3 & 5 \end{vmatrix}</p> <p>\text{Expanding along the first column:}</p> <p>(a-1)\begin{vmatrix} 3-5 & 5 \\ 3 & 5 \end{vmatrix} -(-1)\begin{vmatrix} -1 & 5 \\ 1 & 5 \end{vmatrix} + 1\begin{vmatrix} -1 & 3-5 \\ 1 & 3 \end{vmatrix}</p> <p>=(a-1)((3-5)(5)-5(3)) -(-1)((-1)(5)-5(1)) + (1)((-1)(3)-3(3-5))</p> <p>=(a-1)(-10+15) -(-1)(-5) + (1)(-3+6)</p> <p>=(a-1)(5) -5 + 3</p> <p>=5a-5 -5 +3</p> <p>=5a-7</p> <p>\text{Without more context or additional operations indicated for the other matrices, the calculation of their determinants isn't completed in the image.}</p>
Finding the Inverse of a Matrix
To find the matrix Q given that: \[ \begin{pmatrix} 7 & -1 \\ -2 & 1 \end{pmatrix} \cdot Q = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \] we need to multiply the inverse of the given matrix by the identity matrix I. Since the matrix on the right side of the equation is the identity matrix, essentially we are just looking for the inverse of the given matrix. The inverse matrix Q is found by the formula: \[ Q = A^{-1} = \frac{1}{{\text{det}(A)}} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \] where A is the matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and det(A) is its determinant. First, we calculate the determinant of the matrix A: \[ \text{det}(A) = ad - bc = (7 \cdot 1) - (-2 \cdot -1) = 7 - 2 = 5 \] Then, we create the matrix of minors, but since this is a 2x2 matrix, the matrix of cofactors is just the switch of the positions of a and d and a change of sign for b and c. \[ \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 2 & 7 \end{pmatrix} \] Next, we divide each term of this adjust matrix by the determinant to find the inverse: \[ Q = \frac{1}{5} \cdot \begin{pmatrix} 1 & 1 \\ 2 & 7 \end{pmatrix} = \begin{pmatrix} \frac{1}{5} & \frac{1}{5} \\ \frac{2}{5} & \frac{7}{5} \end{pmatrix} \] So the matrix Q is: \[ Q = \begin{pmatrix} \frac{1}{5} & \frac{1}{5} \\ \frac{2}{5} & \frac{7}{5} \end{pmatrix} \]
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com