Matrices!!

Unit 3: MatricesMatricesMatrix:

A rectangular arrangement of data into rows and columns, identified by capital letters.

DimensionsMatrix Dimensions: Number of rows, m, by the number of columns, n. Read as m by n matrix. Also known as the order of a matrix.

RBC (ROWS BY COLUMNS)

Determine the dimensions of each matrix.

ElementsMatrix Element: Each number in a matrix, identified by its row and column.

Example: amn

Refers to the m-th row and n-th column

ExampleIdentify each element.

a23a12a31a21

Zero Matrix: The additive IDENTITY of matrices. A matrix whose elements are all zeros.

Equal Matrices: Matrices with the same dimensions and equal corresponding elements.

Scalar: A real number factor.

7Adding and Subtracting MatricesWhen matrices have the same dimension you add and subtract them by adding or subtracting each corresponding element. Add or Subtract the following matrices:

A + B

C B

2(B + C)

A matrix equation is an equation in which the variable is a matrix.

You can solve for the variable by adding or subtracting a matrix a matrix to both sides to an equation.

Using Scalar Products

Matrix MultiplicationWhen multiplying matrices A and B, the number of COLUMNS in matrix A MUST be equal to the ROWS in matrix B.

The size of the product is:# rows in A x # columns in B.Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be??

1. x

1x3 and 3x3

1x316Multiplying Matrices Can the following Matrices be multiplied? If so, what dimensions will the product be??

1. x

1x3 and 2x3

no17Can the following Matrices be multiplied? If so, what dimensions will the product be?

1.2.

3. 4.

3x2 and 2x2 3x22x3 and 2x3 no3x1 and 1x2 3x22x3 and 3x2 2x218How to multiply matricesMultiply the elements of each row in the first matrix by the elements in each column of the second matrix

Add the products to get the new element.

Matrix Multiplication

You Try!

You Try!

Solutions:

Equivalent Matrices

DETERMINANT of

MatricesDeterminant of 2 x 2

Determinant of 2 x 2Find the determinant of the following 2x2 matrices:

Determinant of 2 x 2Find the determinant of the following 2x2 matrices:

Step 1: Re-write the first two columns on the right side of the determinant.Determinant of a 3x3 Matrix

STEP 2: Draw a diagonal from each element in the top row diagonally downward. Find the product of the numbers on each diagonal.aeibfgcdhSTEP 3: Then draw a diagonal from each element in the bottom row diagonally upward. Find the product of the numbers on each .

idbhfagecStep 4: Add the products in the first set of diagonals, and then subtract the products from the second set of diagonals.

The value is:

aei + bfg + cdh (gec + hfa + idb)or aei + bfg + cdh gec hfa idb Ex. 2: Evaluate using diagonals.

First, rewrite the first two columns along side the determinant.

Ex. 2: Evaluate using diagonals.

Next, find the values using the diagonals.46000-524Now add the bottom products and subtract the top products. 4 + 60 + 0 0 (-5) 24 = 45. The value of the determinant is 45.ExampleFind the determinant of the following.

Try Some!Find the determinant of the following.

Determinant of a 3x3 Matrix

Determinant of a 3x3 MatrixInverse of

MatricesInverseREMEMBER we denote inverse with a -1 power

So the inverse of matrix A is A-1

Requirement to have an InverseMatrix MUST be square, meaning it has the same number of rows and columns

Matrix MUST NOT have a determinant of zero.Inverse exist?!Does the inverse exist?!?!

Multiplying InverseWhen you Multiply a matrix A times its inverse, the Product is the Identity Matrix.

Identity Matrix is a square matrix where the top left to Bottom right diagonal are all ones, and everything else is a zero

Determine if the following matrices are inverses. 1.

2.

Finding the Inverse of a 2x2

IF

THENFind the inverse of the following matrix.

Use your calculator!2nd Matrix EditPut in your matrix2nd Matrix NAMEGet your matrixX-1

The inverse of a matrix can be used when solving matrix equations.

For Matrices A and B, we can find Matrix X:

IF AX = B

THEN X = A-1B

*Solve for X:

X = A-1BYou Try! Solve Each Matrix Equation:

Solutions:Solve each matrix equation.