2331_Notes_5o3_fill - Math 2331 Linear Algebra Section 5.3...

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Math 2331 Linear Algebra Section 5.3 Cramer's Rule, Inverses and Volumes Cramer's Rule This is a way to solve Ax = b using determinants If det(A) is not zero (i.e. A is square and invertible), 1. Ax = b is solvable for any b 2. Cramer's rule can solve for each component of x individually To find the jth component of b, first replace the jth column of A with the vector b and let's call this matrix j B Then det( ) det( ) j j B x A = Example: Use Cramer's Rule to find the solution to 12 32 6 54 xx −= −+ = 8
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Cramer's Rule also gives us a formula for the inverse of A: Given an nxn matrix A Let be the vector that is the jth column of the nxn identity matrix (a 1 in the jth row) j e Solve j A xe = for all j between 1 and n. The (i,j) entry in the inverse of A is the ith row of x Why? This is the cofactor NOTE THE ORDER !!! ji C
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Cramer's Rule for Inverses 1 det( ) ji ij C A A = Let 11 12 1 21 22 2 12 ... ... ... ... n n nn n n CC C C C C ⎛⎞ ⎜⎟ = ⎝⎠ Then 1 det( ) T C A A = Once again - to find the i,j - entry in the matrix 1 A Cross out row j and column i and find the determinant of the resulting matrix. Multiply
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This note was uploaded on 08/10/2011 for the course MATH 2331 taught by Professor Staff during the Spring '08 term at University of Houston.

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2331_Notes_5o3_fill - Math 2331 Linear Algebra Section 5.3...

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