algebra 2: lesson 5 using matrices to organize data and solve problems

Algebra 2: Lesson 5Algebra 2: Lesson 5Using Matrices to Organize Data

and Solve ProblemsUsing Matrices to Organize Data

and Solve Problems

Warm-up p. 29 #1-51. The Additive _______ Property states that a + (-a) = 0

2. Add 16.5 – (-24.8)

3. Solve: 4 – 3y = 16

4. True or False: By the Commutative Property: j – k = k – j

5. Simplify: -4(x + 1)+ 3(2x – 7)

Warm-up p. 29 #1-51. The Additive Inverse Property states that a + (-a) = 0

2. Add 16.5 – (-24.8) 41.3

3. Solve: 4 – 3y = 16 4 – 16 = 3y; -12 = 3y; -4=y

4. True or False: By the Commutative Property: j – k = k – j

5. Simplify: -4(x + 1)+ 3(2x – 7) -4x – 4 + 6x – 21 = 2x - 25

New Concept: MatrixA matrix is a rectangular array of numbers. The number of rows and columns in a matrix gives the dimensions of the matrix.

A matrix with “r” rows and “c” columns is a matrix of dimension “r × c”.

B = -3 -1 -2 2 -5 -4

Row #1

Row #2

Column #1 Column #3Column #2

Give the dimensions of each matrix.

Give the dimensions of each matrix.

3 × 2 4 × 12× 3

Elements of matricesEach member of the matrix is called an element and has a unique address.

For example, in matrix A, a43 is 5.

What element is located at a23?

Matrix AdditionTo add two matrices of the same dimension, add each element in the first matrix to the element that is in the same location in the second matrix.

Zero MatrixA zero matrix is formed when a matrix is added with its additive inverse matrix.

matrix additive inversematrix

zero matrix

Matrix SubtractionTo subtract two matrices of the same dimensions, A – B, take the opposite, or additive inverse, of B and add it to A.

Matrix SubtractionTo subtract two matrices of the same dimensions, A – B, take the opposite, or additive inverse, of B and add it to A.

Example 2

Find the additive inverse matrix of A.

Add: -A + B

Subtract: A – B.

A = -1 2 -50 -4 3

B = -3 -1 -22 -5 -4

Example 2

A = -1 2 -50 -4 3

B = -3 -1 -22 -5 -4

-A = 1 -2 50 4 -3

-A + B= -2 -3 32 -1 -7

A – B= 2 3 -

3-2 1 7

Ex. 3: Solving a Matrix Equation

Rewrite the equation as a subtraction equation.

Subtract the matrices.

-5 2 -4 3 + X =

-1 -2-5 -4

-5 2 -4 3 = X

-1 -2-5 -4 –

4 -4 -1 -7

X =

Ex 4: Solving for Variables in Matrices

Equal matrices have equal elements in matching locations.

Write equations to make matching locations equal.

a+12 = 18; a = 6

2b = -14; b = -7

23 = a + c; 23 = 6 + c; 23 - 6 = c; c = 17

d = 3b; d = 3(-7); d = -21

a + 12 2b

23 d =

18 -14

a + c 3b

Scalar MultiplicationA scalar is a constant by which a matrix is multiplied.

Scalar Multiplication is analogous to repeated Matrix Addition.

To multiply matrix A by scalar “n”, multiply every element of A by n.

Ex. 4: Scalar Multiplication

-5 2 -4 3

Evaluate: -2M

M =

Ex. 4: Scalar Multiplication

-5 2 -4 3

-2 -10 -4 -8 -6

=

-5 2 -4 3

Evaluate: -2M

M =

Partner Practicepage 32 Lesson Practice a - f

Individual Practicepage 33 #1-29 odd