Math Problem Statement
How to find determinant of 33 matrix , an example
Solution
To find the determinant of a matrix, you can use the cofactor expansion (also called Laplace expansion). The determinant of a matrix is calculated as follows:
Formula for the Determinant of a Matrix
For a matrix of the form:
a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ The determinant of $$A$$, denoted as $$\det(A)$$ or $$|A|$$, is: $$\det(A) = a \cdot \det\begin{pmatrix} e & f \\ h & i \end{pmatrix} - b \cdot \det\begin{pmatrix} d & f \\ g & i \end{pmatrix} + c \cdot \det\begin{pmatrix} d & e \\ g & h \end{pmatrix}$$ Where: - Each $$2 \times 2$$ matrix determinant is calculated as $$\det\begin{pmatrix} p & q \\ r & s \end{pmatrix} = ps - qr$$. ### Example Let's calculate the determinant of the following $$3 \times 3$$ matrix: $$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}$$ #### Step 1: Break it down into smaller $$2 \times 2$$ matrices $$\det(A) = 1 \cdot \det\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} - 2 \cdot \det\begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} + 3 \cdot \det\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix}$$ #### Step 2: Compute the $$2 \times 2$$ determinants $$\det\begin{pmatrix} 5 & 6 \\ 8 & 9 \end{pmatrix} = (5 \cdot 9) - (6 \cdot 8) = 45 - 48 = -3$$ $$\det\begin{pmatrix} 4 & 6 \\ 7 & 9 \end{pmatrix} = (4 \cdot 9) - (6 \cdot 7) = 36 - 42 = -6$$ $$\det\begin{pmatrix} 4 & 5 \\ 7 & 8 \end{pmatrix} = (4 \cdot 8) - (5 \cdot 7) = 32 - 35 = -3$$ #### Step 3: Substitute the values into the formula $$\det(A) = 1 \cdot (-3) - 2 \cdot (-6) + 3 \cdot (-3)$$ $$\det(A) = -3 + 12 - 9$$ $$\det(A) = 0$$ So, the determinant of matrix $$A$$ is $$0$$. ### Summary: The determinant of the $$3 \times 3$$ matrix $$A$$ is $$0$$, which means that the matrix is **singular**, and thus it does not have an inverse. --- **Would you like a detailed explanation on any specific part of this process?** Feel free to ask if you have any questions! --- ### Related questions: 1. How do you calculate the inverse of a $$3 \times 3$$ matrix? 2. What does it mean for a matrix to be singular? 3. How does the determinant relate to the volume of a parallelepiped? 4. How do you find the determinant of a $$2 \times 2$$ matrix? 5. Can the determinant be negative or zero? **Tip**: If you encounter a matrix with a row or column of all zeros, the determinant will always be zero. This is a quick check to see if the matrix is singular!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Cofactor Expansion
Formulas
For a 3x3 matrix A = [[a, b, c], [d, e, f], [g, h, i]], det(A) = a * det([[e, f], [h, i]]) - b * det([[d, f], [g, i]]) + c * det([[d, e], [g, h]])
Theorems
Cofactor Expansion
Properties of Determinants
Suitable Grade Level
Grades 10-12