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Lecture 28 - Mar 16

# Lecture 28 - Mar 16 - Monday March 16 Lecture 28...

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Monday March 16 Lecture 28 : Determinants and invertibility . (Refers to sections 5.2) Objectives : 1. Recognize that det( AB ) = det( A )det( B ). 2. Recognize that det( cA ) = c n det( A ). 3. Recognize that a matrix A is invertible iff det( A ) 0. 4. Recognize that det( A -1 ) = 1 / det( A ). 5. Define the adjoint of a matrix. 6. Find the inverse of a matrix by using the adjoint of the matrix. 7. Apply Cramers rule to find to solve a system of linear equations. 28.1 Theorem If A and B are two square matrices of same dimension then det( AB ) = det( A )det( B ) . (Proof is omitted) 28.2 Theorem If A is an n × n matrix and c is a scalar, then the det( cA ) = c n det( A ) . 28.3 Theorem. 1. If the square matrix A has an inverse then det( A ) 0 . 2. If A is a matrix such that det( A ) 0 then A is invertible. Proof of " If the square matrix A has an inverse then det( A ) 0": o Suppose that A is invertible. o Then A is a product of elementary matrices, A = E 1 E 2 E 3 ... E n . o Then det( A ) = det( E 1 )det( E 2 )det( E 3 )...det( E n ). Why? o The determinant of an elementary matrix is either 1, a non-zero scalar or 1, depending on whether it is an elementary matrix of type I, II or III, respectively. o The determinant of an elementary matrix is never zero. Why? o Thus det( A ) 0. Proof of " If A is a matrix such that det( A ) 0 then A is invertible ":

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o Suppose now that det( A ) 0. o Then det( A RREF ) is not zero. Why? (det( A RREF ) is the product of det( A ) and the determinant of elementary matrices.) o Thus A RREF does not have a row or zeros.
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