Determinants and Inverses

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Determinants and Inverses : 

Determinants and Inverses How do you calculate determinants and inverses of 2 x 2 and 3 x 3 matrices?

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A multiplicative identity matrix is an n x n matrix with 1s along the main diagonal and 0s elsewhere. 2 x 2 Multiplicative Identity 3 x 3 Multiplicative Identity

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Remember from a previous lesson: If ab = 1, then b is the multiplicative inverse of a so… If A and X are square matrices and AX = XA = I, where I is the identity matrix, then X is the multiplicative inverse of A–written A-1.

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To show that 2 matrices are inverses of each other, show AB = I or BA = I. For example, show that B is the multiplicative inverse of A: Show that AB = I.

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Every square matrix with real number elements has a real number determinant. Determinants help you find inverses. To find the determinant of a 2 x 2 matrix, use the following formula: Notice: The brackets change to lines to represent Determinants.

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For example: 6(-3) – 2(-1) =-18 + 2 =-16 6(2) – 1(-3) =12 + 3 =15 a(a) – 3(2) =a2 - 6

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To find the determinants of a 3 x 3 matrix, you will first need to recopy the first two columns. 1. Multiply the downward diagonals. 2. Multiply the upward diagonals. 3. Add the results. 4. Subtract the top sum from the bottom sum.

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Find the determinant of the matrix below: 1. Multiply the upward diagonals. We get 2, 0, 0. 2. Multiply the downward diagonals. We get 6, -12, and 0. 3. Find the sums. Sum = 2 Sum = -6 4. Subtract the top sum from the bottom sum. Det = -6 – 2 = -8!!

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Another example: Find the determinant of Recopy the first 2 columns. Multiply the upward diagonals. Multiply the downward diagonals. Find the sums. Subtract the sums. We get -9, 16, and 0. We get 2, -6, and 0. Sum = 7 Sum = -8 Determinant = -8 – 7 = -15!!