The Determinant of a Matrix Note: The determinant of a matrix can be positive, zero, or negative. Chapter 3 Determinants

3-2 Notes: Sign pattern for cofactors

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3-5 The determinant of a matrix of order 3: Add these three products. Subtract these three products.

3-6 Upper triangular matrix: Lower triangular matrix: Diagonal matrix: All the entries below the main diagonal are zeros. All the entries above the main diagonal are zeros. All the entries above and below the main diagonal are zeros. Ex: upper triangularlower triangulardiagonal

3-7 A row-echelon form of a square matrix is always upper triangular.

Evaluation of a Determinant Using Elementary Operations

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3-10 Determinants and Elementary Column Operations: The elementary row operations can be replaced by the column operations and two matrices are called column-equivalent if one can be obtained form the other by elementary column operations.

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Properties of Determinants Notes: (1) (2)

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Applications of Determinants Matrix of cofactors of A: Adjoint matrix of A:

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