Matrix Algebra

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

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.

Finding the Inverse of a 2x2 matrix 17/11/2018 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

Example Find Inverse of A 17/11/2018 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

Find the inverses and check them 17/11/2018 Find the inverses and check them inverse matrix

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

Why will it help us solve equations? 17/11/2018 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

Solving systems of equations 17/11/2018 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

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).

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  

Your Turn solve the following 17/11/2018 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 = -0.16 y = 0.2 Answer x = -2 y = -5 inverse matrix