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Determinant The numerical value of a square array of numbers that can be used to solve systems of equations with matrices. Second-Order Determinant (of a 2 x 2 Matrix) The word with the matrix symbol The Determinant symbol (Criss-Cross & multiply then subtract)

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Example 1

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nth-Order Determinant The determinant of any n x n (square) matrix Third-Order Determinant (of a 3 x 3 Matrix) Determinant of what’s left Signs alternate, beginning with minus Then simplify

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Example 2 Select any row & column, then calculate the determinant with expansion of the minors

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Identity Matrix for Multiplication A square matrix, that when multiplied with another square matrix, results in a matrix with no change. The square matrix always consists of 1’s on the diagonal beginning with the first element; the remaining elements are zeros.

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Inverse Matrix (A -1 ) A matrix that when multiplied by another matrix results in the identity matrix. Note: Not all matrices have an inverse – If the determinant of the original matrix has a value of zero, A -1 does not exist

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Example 3 Determine if a matrix exists (Det ≠ 0) 4 2 3 -4 Switch places Same place Switch Signs The inverse is used to solve systems of equations with matrices.

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Calculate the determinant: Determine the inverse: Multiply each side of the matrix equation by the inverse & solve:

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Example 5 Define the variables: Let x = amount of 10% bond Let y = amount of 6% bond Two variables, two equations: x + y = 10,500 Simplify; no decimals. Write matrix equation:Determine the inverse: 1(-5) – 3(1) = -8 3x – 5y = 0

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HW: Page 102

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Inverse Matrices (2 x 2) How to find the inverse of a 2x2 matrix.

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