1. Matrix
Algebra

2. Multiplying a matrix by the identity gives the matrix back again.
What is AI?
What is IA?
identity matrix
an n  n matrix with ones on the main diagonal and zeros elsewhere

3. Can we find a matrix to multiply the first matrix by to get the identity?
Let A be an n n matrix. If there exists a matrix B such that AB = BA = I then we call this matrix the inverse of A and denote it A-1.

4. Finding the Inverse of a 2x2 matrix
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Finding the Inverse of a 2x2 matrix
Step-1 First find what is called the Determinant
This is calculated as ad-bc
Step-2 Then swap the elements in the leading diagonal
Step-3 Then negate the other elements
Step-4 Then multiply the Matrix by 1/determinant
inverse matrix

5. Example Find Inverse of A
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Example Find Inverse of A
Step 1 – Calc Determinant
Determinant (ad-cb) = 4x3-8x1 = 4
Step 2 – Swap Elements on leading diagonal
Step 3 – negate the other elements
Step 4 – multiply by 1/determinant
inverse matrix

6. Find the inverses and check them
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Find the inverses and check them
inverse matrix

7. We can use A-1 to solve a system of equations
To see how, we can re-write a system of equations as matrices.
coefficient matrix
variable matrix
constant matrix

8. Why will it help us solve equations?
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Why will it help us solve equations?
Because if we can express a system of equations in the form
Then we can multiply both sides by the inverse matrix
And we can then know the values of X because
inverse matrix

9. Solving systems of equations
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Solving systems of equations
We can use our 2x2 matrices to express these systems of equations
Becomes in matrix form
constants from the right hand side
constants from the left hand side
UNKNOWNS
X ~ x1
Y ~ x2
inverse matrix

10. Your calculator can compute inverses and determinants of matrices.
For the inverse:
Step 1: enter the matrix into the calculator
Step 2: Pull up matrix entered onto the screen
Step 3: hit the x^-1 button and enter (this will give you the inverse matrix).

11. Now using the previous problem we will allow the calculator to do the work.
Enter this as matrix A in the calc
Enter this as Matrix B in the calc

12. Your Turn solve the following
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Your Turn solve the following
3x +4y = 5
5x = 7-6y
x+7y = 1.24
3y -x = 0.76
8x = 3y -1
x+y =-7
Answer x = -1 y = 2
Answer x = y = 0.2
Answer x = -2 y = -5
inverse matrix