# [[Determinant of a 3x3 Matrix]] (Mathematics > Linear Algebra) ## Definition The **determinant of a $3 \times 3$ matrix** $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ is a scalar value that provides information about the matrix's invertibility, volume scaling factor, and orientation in space. The determinant of $A$ is given by: $ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). $ ## Key Concepts - **Minors and Cofactors**: The terms $(ei - fh)$, $(di - fg)$, and $(dh - eg)$ are minors of $A$ associated with the elements $a$, $b$, and $c$, respectively. - **Sign Pattern**: The determinant expansion uses alternating signs ($+, -, +$) for the cofactors. - **Invertibility**: $A$ is invertible if and only if $\det(A) \neq 0$. ## Important Properties 1. **Linear in Rows/Columns**: The determinant is a linear function of each row and column. 2. **Swapping Rows/Columns**: Swapping two rows or columns changes the sign of the determinant. 3. **Row or Column of Zeros**: If any row or column of $A$ is all zeros, then $\det(A) = 0$. ## Essential Formulas - **Determinant Formula for $3 \times 3$ Matrix**: $ \det(A) = a(ei - fh) - b(di - fg) + c(dh - eg). $ ## Core Examples 1. **Example Calculation**: Given $A = \begin{bmatrix} 2 & 3 & 1 \\ 4 & -1 & 5 \\ 7 & 2 & 0 \end{bmatrix}$, calculate $\det(A)$: $ \det(A) = 2((-1)(0) - (5)(2)) - 3((4)(0) - (5)(7)) + 1((4)(2) - (-1)(7)) $ $ = 2(0 - 10) - 3(0 - 35) + 1(8 + 7) = -20 + 105 + 15 = 100. $ 2. **Geometric Interpretation**: The absolute value $|\det(A)|$ represents the volume scaling factor for transformations in $\mathbb{R}^3$ by matrix $A$. ## Related Theorems/Rules - **Cofactor Expansion**: The determinant can be expanded along any row or column using minors and cofactors. - **Properties of Determinants**: For example, $\det(AB) = \det(A) \det(B)$ for square matrices $A$ and $B$. ## Common Pitfalls - Forgetting the alternating signs in the cofactor expansion. - Calculating minors incorrectly by omitting or misidentifying elements in submatrices. ## Related Topics - [[Matrices and Their Dimensions]] - [[Special Types of Matrices]] - [[Minor of a Matrix Element]] - [[Matrix Multiplication]] - [[Matrix-Vector Multiplication]] - [[Linear Transformation]] - [[Transpose of a Matrix]] - [[Scalar Multiplication of Matrices]] - [[Linear Combinations of Vectors]] - [[Area of a Parallelogram Using Determinants]] - [[Cofactor Expansion]] - [[Matrix Inverses and Determinants]] - [[Properties of Determinants]] ## Quick Review Questions 1. Calculate the determinant of $A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \end{bmatrix}$. 2. Why does a $3 \times 3$ matrix with a row of zeros have a determinant of zero? *** ![[Determinant_of_a_3x3_Matrix_visualization]]