ENTER THE MATRIX THE BASICS WHAT IS A

ENTER THE MATRIX THE BASICS…

WHAT IS A MATRIX? • Row column

AUGMENTED MATRICES •

Row Operations: 1) Any two rows can be interchanged 2) The elements in any row can be multiplied by a nonzero number 3) The elements in any row can be multiplied by a nonzero number and the result can be added to any other row Two matrices are said to be row equivalent if one can be obtained from the other using row operations.

GAUSSIAN ELIMINATION

EX 3. Solve the system using Gaussian Elimination with back substitution: 2 x + y – z = 1 2 x – 3 y + z = 1 x+y+z=4

MATRIX OPERATIONS • Only matrices of the same order can be added or subtracted • Matrices can only be multiplied if the number of columns in the first is the same as the number of rows in the second.

• IDENTITY CRISIS?

Any matrix multiplied by its identity results in itself: AI = A, IA = A Likewise, any matrix multiplied by its inverse is the identity: A-1 A = I Which leads us to…

THE INVERSE OF A SQUARE MATRIX • Only square matrices have inverses • Inverse matrices when multiplied together always result in the identity • You can use row operations to get the inverse, but for 2 x 2’s there is a short cut!

2 X 2 INVERSE •

DETERMINANTS • Finding the inverse of a 2 x 2 actually uses a determinant • Determinants of 2 x 2’s and 3 x 3’s are actually pretty easy to find

WHY MATRICES? As we saw with Gaussian Elimination, matrices can be used to solve systems. There are other methods other than just Guass… We can also solve systems using matrices by solving a matrix equation as well as Cramer’s Rule!

MATRIX EQUATIONS Ex. 7 Solve the system using a matrix equation. 5 x – 3 y =12 2 x + 8 y = -7

CRAMER’S RULE (2 X 2) •

CRAMER’S RULE (3 X 3) •

EX 8. Use Cramer’s Rule to solve the following system. 3 x – 2 y = 7 -8 x – y = 4

CALCULATOR MAGIC! Graphing calculators have the ability to make your matrix experience a more pleasant one…