Supplement #2 Matrix Inversion

MATH HEMATICS S TUTOR RIAL INTER RNATIONAL UNDE ERGRADU UATE PRO OGRAMM ME By Gillang (gum milangaryo [email protected])

SUPP PLEMEN NT #2

MAT TRIX INVE ERSIO ON WIT TH ER RO

“When somebody s challenge you, fight back. Be brutal, b be ttough!” mp – – D. Trum

MATR RIX INVER RSION Matrix x inversion n conceptt is usefull to solve several prroblems s such as so olving system ms of n-un nknowns or input-u utput ana alysis. Ste eps that y you have to t remem mber in matrix m inve ersion. 1. Calculate e the deterrminant e the mino ors 2. Calculate 3. Find the cofactors 4. TRANSPO OSE the cofactor c m matrix 5. Inversed Matrix is

1 | |

| |.

Example: 1 2

1 1

We can use the classic steps to find the inverse of Matrix A above. First of all, we shall calculate the determinant. | |

1

1

1 2

3 Next, we calculate the minors. 1 1

2 1

Then, we find the cofactors. (Remember that C=(-1)i+jMij) 1 1

2 1

After that, we transpose the cofactor matrix. | |

1 2

1 1

The inverse of matrix A is 1 | |

| |.

1 1 1 3 2 1 1/3 1/3 2/3 1/3

MATRIX INVERSION WITH ERO Now, recall Elementary Rows Operation (ERO) in supplement #1. We can use ERO to solve matrix inversion. The two steps are given below. 1. Form a two-side matrix where the left hand side is the matrix that will be inversed and the right hand side is identity matrix. 2. Use steps in ERO to transform the left hand side matrix into an identity matrix. The right hand side matrix will be the inverse of the matrix. Example: 1 2

1 1

First, we form the matrix as suggested in step 1. 1 2

1 1 || 1 0

0 1

Next, we can apply steps in ERO to find the matrix inversion as suggested in step 2. R1+R2 3 2

0 1 || 1 0

1 1

1/3 (R1) 1 2

0 1/3 || 1 0

1/3 1

R2-2R1 1 0

0 1/3 1/3 || 1 2/3 1/3

-1(R2) 1 0

0 1/3 || 1 2/3

1/3 1/3

We can see that the left hand side of the matrix form an identity matrix, hence, the right hand side is the inverse of the matrix. 1 0

0 1/3 || 1 2/3

1/3 1/3

1/3 2/3

1/3 1/3

The inverse of matrix A is

Now imagine inversing a 4x4 matrix ! Using ERO is simpler right? (I strongly recommend using matrix inversion for 2x2 & 3x3 matrices)

TIPS In the second part of the discussion in matrix we learned how to solve system of n-unknowns e.g. x & y or x, y, & z. This is my personal preference of the methods used in solving problems of n-unknowns. For me, it’s the best in the use of our time and mind. 1. To solve system of 2-unknowns, I prefer substitution method or Cramer’s rule. 2. To solve system of 3-unknowns, I prefer Cramer’s rule. 3. To solve system of 4- or higher-unknowns, I prefer ERO. The choice is yours!