To determine the determinant of a matrix, you need to follow specific steps depending on the matrix's size. Here's a brief guide:

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1. Determinant of a 2×2 Matrix

For a matrix: $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ The determinant is given by: $\text{det}(A) = ad - bc$

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2. Determinant of a 3×3 Matrix

For a matrix: $A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$ The determinant is computed using the rule of Sarrus or cofactor expansion: $\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$

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3. Determinant of an $n×n$ Matrix (Higher Dimensions)

For larger matrices ($n>3$):

• Use cofactor expansion along a row or column.
• Alternatively, apply row reduction to bring the matrix to triangular form, then multiply the diagonal entries.

The determinant involves recursively breaking the matrix into smaller matrices (minors) until 2×2 determinants are reached.

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If you provide a specific matrix, I can compute the determinant for you step by step!

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Relative Questions:

1. What are the practical uses of matrix determinants?
2. How does cofactor expansion work for higher dimensions?
3. What is the relationship between a matrix being invertible and its determinant?
4. How can the determinant help in solving systems of linear equations?
5. What are the properties of determinants in row and column operations?

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Tip:

For large matrices, software