Unit 2, Lesson 5
What is the determinant of a 3 by 3 matrix?
(a11a22a33 + a21a32a13 + a12a23a31) - (a31a22a13 + a11a32a23 + a12a21a33)
How to Solve AX = B Using an Inverse
1. Begin by calculating the determinant for the first matrix 2. Use the determinant to calculate the inverse 3. Use the inverse to solve for X
How to Find the Inverse of a Matrix
1. Begin by finding the determinant; inverse exists if determinant isn't equal to 0 2. Use inverse equation 3. Substitute in the determinant to calculate the inverse matrix
How to Solve a System of Three Equations Using an Inverse Matrix
1. Write the system as an inverse equation 2. Find the inverse of matrix A using a graphing calculator or online matrix calculator 3. Multiply the inverse of matrix A by the constant matrix to determine the variable matrix values 4. The answer is expressed as an ordered triple
I is a matrix that has
1s in the main diagonal & 0s as the remaining elements
The inverse of matrix A can be used to solve matrix equations in the form
AX = B
Determinant for a 2 By 2 Matrix
Determinant is equal to the difference between the products of the diagonal elements det A = a11(a22) - a12(a21)
How to Solve AX = B
Use Associative Property of Multiplication & multiplicative inverse definition AX = B A-1(AX) = (A-1)B (A-1(A))X = (A-1)B IX = (A-1)B I is the multiplicative identity, so further simplify X = (A-1)B
Given a square matrix A, which has a determinant not equal to zero, then
a matrix exists, with the same dimensions, such that the product of the two matrices is equal to I A(A-1) = (A-1)A = I
nonsingular matrix
a matrix that has a nonzero determinantumblr.com
If a coefficient matrix has an inverse, then the system has
a unique solution
What happens if one or more of the equations doesn't have a variable?
a zero is placed in matrix A to represent the coefficient
square matrix
any matrix with the same number of columns as rows
How is the determinant of a 3 by 3 matrix found?
by taking the sum of the diagonal products in one direction and subtracting them from the sum of the diagonal products in the other direction
minor of an element
determinant of order (n - 1); can be found by deleting row & column containing element
To verify if two 3 by 3 matrices are inverses,
find their product
If a coefficient matrix doesn't have an inverse, then the system has
no solution or infinite solutions
The coefficient matrix, variable matrix, & constant matrix are combined to
produce the matrix equation that represents the systems of equations
coefficient matrix
the A matrix in an AX = B matrix equation; composed of the coefficients of the terms in each equation
constant matrix
the B matrix in an AX = B matrix equation; composed of the constants in the system of equations
To verify that a given matrix, B, is the multiplicative inverse of matrix A,
the product of A & B must be I
In order to use matrices to solve a system of equations,
the system must be expressed as a matrix equation
How do you check for a unique solution using matrices?
use the determinant of the coefficient matrix; if the determinant is zero, the system has no unique solution (because a determinant of zero means there's no inverse, which means there's no unique solution)
determinants
used to determine if there's an inverse of a matrix & to solve systems of linear equations
When does an inverse for a matrix exist?
when the matrix has a determinant that's not equal to zero
