Intro To Linear Algebra 3.1 Inverse, Elementary, and Determinants Matrices PDF

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Summary of Lectures 3-5 from Intro To Linear Algebra taught by James Carlo October 3, 2018 for Math 261 during Fall Term 2018....

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3.1 Inverse, Elementary, and Determinants Matrices Properties of Inverse Matrices: Given AX=B (where A is a matrix, X is a vector, and B is the product of A and X): o (A-1*A)X=A-1*B (where A-1 is the inverse matrix of matrix A) o I*X=A-1B (where I is the identity matrix) o X=A-1*B  Elementary Matrices: is always a square matrix by applying one row operation to the identity matrix of the same size. Given Original Matrix B: 1 1 R1 0 2 1 R2 1 2 1 0 R3 ex) Row swapping aka Permutation Operation: R1 swapped with R2 in Original Matrix B Note: The notation for a permutation/swap is pa,b where ‘a’ is the first 1 2 1 row that is R1 0 swapped with ‘b’, the second row. 1 1 R2 1 0 R3 2 =p1,2 o ex) Multiplying a row by a constant Operation: 3*(R1) in Original Matrix B Note: The notation for multiplying a row by a constant operation is 0 1 1 E(a*Row R1 3 where ‘a’ is the constant being multiplied, and ‘Row’ is the 6 3 operation R2 2 row that is being multiplied by the constant. 1 0 R3 =E(3*R1) o ex) Adding one row to another row Operation: R1+R2 in Original Matrix B 

R1 R2 R3

1 1 2

3 2 1

2 1 0

Note: The notation for adding a row to another row is E(a+b) where ‘a’ is the first row that is added to another row, ‘b’.

=E(R1+R2) Invertible Matrices  Two matrices paired together in augmented matrix form means that whatever happens to one side (to one matrix) of the line, must happen to the matrix on the other side of the line.  The Reduced Row-Echelon Form of matrix A is I, the identity matrix. Therefore, I=U*A where U is the product of the elementary matrices (Ek-1) and A=U-1. A/ U-1 as a product of the elementary matrices is therefore A= Ek-1*…E2-1*E1-1. Properties of Invertible Matrices: o The Reduced Row-Echelon Form of the invertible matrix is the Identity matrix



ex) matrix A

Operations:

reduced row-echelon form of A

E1=p1,2 E2=E(2*R2+R3

0 1 0

1 1 -2

0 0 1

)

1 0 0

0 1 0

0 0 1

E3=(R1-R2)

o The Invertible matrix is square  ex) possible nxn dimensions:  3x3  2x2  1x1 (unless the value within the 1x1 matrix is equal to 0 because the determinant would be equal to 1/0 which is undefined. ) o If A is invertible it can be rewritten as a product of elementary matrices ex) A=E1*E2*E3 Properties of Determinants: o Switching two rows in matrix A is results in matrix B means the determinant of B is equal to the opposite sign of the determinant of A: det(B)=-det(A) o Multiplying a row in matrix A by a nonzero number, or scalar ‘k’ results in matrix B. The determinant of B will equal k times the determinant of A. o Replacing a row by a multiple of another row added to the first row, if A and B are square (nxn) matrices with k scalars such that B=k*A, then the determinant of B will equal k to the power of n times determinant of A: det(B)=kndet(A). th  The ij minor of a Matrix: o If A is an nxn matrix, then the ijth minor of A is the determinant of the (n-1)x(n-1) matrix which results from deleting the ith row and the jth column of A.  ex) If A is a 3x3 matrix, the ijth minor of A is minor(A)ij which is the determinant when the ith row and jth column is deleted and leaves a 2x2 matrix.  Cofactors: o If A is an nxn matrix with ijth cofactor, the cofactor of A at ij will equal negative one to the i+j power times the minor of A at i and j: cof(A)ij=(-1)i+j*minor(A)ij o Therefore, the determinant of a matrix called A, where aij is a component of matrix A, is equal to: det(A)=ai1*cof(A)i1+ai2*cof(A)i2+ai3*cof(A)i3 o If we let A be a 3x3 matrix, the determinant of A is calculated by picking a row (or column) and taking the product of each entry in the row (or column) with its cofactor and adding these products together.  This is also known as expanding the ith row (or column) when applied to the ith row= Laplace expansion/ Cofactor expansion of a Matrix  Note: expanding the nxn matrix along any row or column always gives the same answer, which is the determinant.



Triangular Matrices: where the determinant of a matrix called A is the product of the pivot points, called diagonal entries if there are only zeroes above or below the diagonal entries.

ex)

1 0 0

2 6 0

4 3 3

Note: Since the the matrix has nxn dimensions and a triangular set of zeroes under its pivot points, it is a triangular matrix....