Adding & Subtracting Matrices

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Matrices

In this Workshop you will:

• Learn the definition of a matrix.• Identify elements of a matrix.• Learn how to add matrices.• Learn how to subtract matrices.• Learn how to solve variable expressions

using addition and subtraction of matrices.

Definition of a Matrix

• A matrix is a rectangular or square array of elements arranged in rows and columns.

• Numbers in a matrix are called its entries or elements.

NOTE: Entries may be numerical or variable expressions.

m= number of rows

n= number of columns

amn= any element in row m and column n

m×n= dimension of a matrix

mnmm

n

n

aaa

aaa

aaa

A

21

22221

11211

Elements of a Matrix

Addition• If two or more matrices have the same order (same

number of rows and columns) matrices may be added to their corresponding entries.

mnmm

n

n

mnmm

n

n

bbb

bbb

bbb

aaa

aaa

aaa

BA

21

22221

11211

21

22221

11211

mnmnmmmm

nn

nn

bababa

bababa

bababa

2211

2222222121

1112121111

Example 1: Find A + B

376

928

541

A

157

843

269

B

157

843

269

376

928

541

BA

135776

894238

256491

41213

17611

71010

Solution:

Subtraction• Like in addition, matrices that have the same order may

be subtracted from their corresponding entries.

mnmm

n

n

mnmm

n

n

bbb

bbb

bbb

aaa

aaa

aaa

BA

21

22221

11211

21

22221

11211

mnmnmmmm

nn

nn

bababa

bababa

bababa

2211

2222222121

1112121111

Example 2: Find A - B

43701925

28613251A

1985123

1291416

43701925

28613251BA

1985123

1291416B

)19()43()8()70()51()19()23()25(

)12()28()9()61()14()32()16()51(

24787048

16701835

Solution:

Variables in Matrices

• Use A + B = C to solve for x, y, and z.

zx

yxA

42

2

11

yB

50

73C

CBA

50

73

2

1142

yzx

yx

50

73

2

1412

zyx

yx

Solution:

•Add elements in matrix A with elements in Matrix B

•Rewrite elements in matrix A+B as variable expressions

52 z3z

2x

312 x42 x

714 y

2y84 y

Each Element in Matrix A+B corresponds to an element in matrix C. Therefore, the solutions can be found by treating each element as a linear equation and solving for the indicated variable.

NOTE: Once x and y are found, there is no need to use A21

+B21 = C21 to solve.

A11+ B11 = C11 A12+ B12 = C12 A22+ B22 = C22

Matrices Links

• Adding and Subtracting Matrices Handout• Matrices Handout• Matrices Quiz• Multiplying and Dividing Matrices Workshop• Multiplying Matrices Handout• Gauss-Jordan Workshop• Gauss-Jordan Method Handout• Inverse Matrix Handout