Determinant of a 3x3 Matrix

Math Problem Statement

Find the determinant of the matrix: \[ \begin{bmatrix} 1 & 0 & 7 \\ 4 & 5 & 9 \\ 1 & 6 & 2 \end{bmatrix} \]

The image shows a matrix, and it looks like you're being asked to find its determinant. The matrix provided is:

1 & 0 & 7 \\ 4 & 5 & 9 \\ 1 & 6 & 2 \end{bmatrix}$$ Now, let's calculate the determinant of this 3x3 matrix using the formula: $$\text{det}(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ Where the matrix elements are: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$ In your matrix: - $$a = 1$$, $$b = 0$$, $$c = 7$$ - $$d = 4$$, $$e = 5$$, $$f = 9$$ - $$g = 1$$, $$h = 6$$, $$i = 2$$ ### Step-by-step solution: 1. Compute the first term: $$a(ei - fh)$$ $$1 \times (5 \times 2 - 9 \times 6) = 1 \times (10 - 54) = 1 \times (-44) = -44$$ 2. Compute the second term: $$-b(di - fg)$$ $$0 \times (4 \times 2 - 9 \times 1) = 0 \times (8 - 9) = 0$$ 3. Compute the third term: $$c(dh - eg)$$ $$7 \times (4 \times 6 - 5 \times 1) = 7 \times (24 - 5) = 7 \times 19 = 133$$ Now sum the results: $$\text{det}(A) = -44 + 0 + 133 = 89$$ Thus, the determinant of the matrix is **89**. Let me know if you'd like more details on the steps or further explanation! ### Questions: 1. How does the determinant help in understanding matrix properties? 2. What happens if the determinant of a matrix is zero? 3. Can you explain the cofactor expansion method for finding a determinant? 4. How is the determinant related to matrix inverses? 5. What are some practical applications of matrix determinants? **Tip:** When calculating determinants for large matrices, it's useful to first check if any rows or columns can be simplified (e.g., zeros) to make the calculation easier.

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Linear Algebra
Matrix Determinant

det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
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