Math Problem Statement
determinant
What is the determinant of the given matrix?
Solution
The matrix in the image is:
2 & 1 & -3 \\ 1 & 1 & 0 \\ 0 & 4 & -1 \end{bmatrix}$$ Now let's compute the determinant of this 3x3 matrix. The determinant $$\text{det}(A)$$ for a 3x3 matrix: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ is calculated as: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For the given matrix, we have: $$\text{det}(A) = 2 \cdot (1 \cdot -1 - 0 \cdot 4) - 1 \cdot (1 \cdot -1 - 0 \cdot 0) + (-3) \cdot (1 \cdot 4 - 1 \cdot 0)$$ $$= 2 \cdot (-1) - 1 \cdot (-1) + (-3) \cdot 4$$ $$= -2 + 1 - 12 = -13$$ So, the determinant of the matrix is $$-13$$. Would you like further details on this computation or have any other questions? Here are some related topics: 1. How is the determinant of a matrix used in solving systems of equations? 2. What are some properties of determinants in larger matrices? 3. How does the determinant change when rows or columns are swapped? 4. How does a determinant help in finding the inverse of a matrix? 5. What is the significance of a zero determinant? **Tip:** A matrix with a zero determinant is called singular, meaning it does not have an inverse.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrices
Formulas
det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Determinant of a 3x3 matrix
Suitable Grade Level
Grades 10-12