Inverses
and
Solving
Systems
4-5
4-5 Matrix
Matrix
Inverses
and
Solving
Systems
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Algebra
Holt
Algebra
22
4-5 Matrix Inverses and Solving Systems
Warm Up
Multiple the matrices.
1.
Find the determinant.
2.
Holt Algebra 2
–1
3.
0
4-5 Matrix Inverses and Solving Systems
Objectives
Determine whether a matrix has an
inverse.
Solve systems of equations using
inverse matrices.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Remember!
The identity matrix I has 1’s on the main
diagonal and 0’s everywhere else.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Example 1A: Determining Whether Two Matrices Are
Inverses by using your graphing calculator
Determine whether the two given matrices
are inverses.
The product is the identity
matrix I, so the matrices
are inverses.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Example 1B: Determining Whether Two Matrices Are
Inverses by using your graphing calculator
Determine whether the two given matrices
are inverses.
Neither product is I, so the
matrices are not inverses.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Check It Out! Example 1
Determine whether the given matrices are
inverses.
The product is the
identity matrix I, so the
matrices are inverses.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Example 2: Finding the Inverse of a Matrix by using
your graphing calculator
Find the inverse of the matrix if it is defined.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Check It Out! Example 3
Find the inverse of
Holt Algebra 2
, if it is defined.
4-5 Matrix Inverses and Solving Systems
You can use the inverse of a matrix to solve a system
of equations. This process is similar to solving an
equation such as 5x = 20 by multiplying
each side by
, the multiplicative inverse of 5.
To solve systems of equations with the inverse, you
first write the matrix equation AX = B, where A is
the coefficient matrix, X is the variable matrix,
and B is the constant matrix.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
The matrix equation representing
Holt Algebra 2
is shown.
4-5 Matrix Inverses and Solving Systems
To solve AX = B, multiply both sides by the inverse A-1.
A-1AX = A-1B
IX = A-1B
X = A-1B
Holt Algebra 2
The product of A-1 and A is I.
4-5 Matrix Inverses and Solving Systems
Example 4: Solving Systems Using Inverse Matrices
Write the matrix equation for the system and solve.
Step 1 Set up the matrix equation.
A
X =
B
Write: coefficient matrix  variable
matrix = constant matrix.
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Example 4 Continued
Step 2: Solve by plugging [A]-1 B in to your
graphing calculator
 x  5
Solution is     
 y   2
Holt Algebra 2
.
4-5 Matrix Inverses and Solving Systems
Example 5
3 x  2 y  z  1
Write the matrix equation for
and solve.
x  3 y  z  10
2 x  y  3 z  3
Holt Algebra 2
4-5 Matrix Inverses and Solving Systems
Lesson Quiz: Part I
1. Determine whether
inverses.
and
yes
2. Find the inverse of
Holt Algebra 2
, if it exists.
are
4-5 Matrix Inverses and Solving Systems
Lesson Quiz: Part II
Write the matrix equation and solve.
3.
4. Decode using
"Find the inverse."
Holt Algebra 2
.