A =

a11 ,…, a1n

a21 ,…, a2n

… … … …am1 ,…, amn

= Aij{ }

A matrix is any doubly subscripted array of elements arranged in rows and columns.

[1 x n] matrix

 

[ ] { }jn aaaaA ,, 2 1 =…

{ }i

m

a

a

a

a

A 2

1

=

…=

[m x 1] matrix

B =

5 4 7

3 6 1

2 1 3

Same number of rows and columns

The Ident i t y

I =

1 0 0 0

0 1 0 0

0 0 1 0

0 0 0 1

Square matrix with ones on the diagonal and zeros elsewhere.

A' =

a11 a21 ,…, am1

a12 a22 ,…, am 2

… … … … …a1n a2n ,…, amn

Rows become columns and columns become rows

A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B is defined by: 

Cij{ } = Aij{ } + Bij{ }

Note: all three matrices are of the same dimension

A = a11 a12

a21 a22

B = b11 b12

b21 b22

C = a11 + b11 a12 + b12

a21 + b21 a 22 + b22

If

and

then

A + B = 3 4

5 6

+

1 2

3 4

=

4 6

8 10

= C

C = A - BIs defined by

Cij{ } = Aij{ } − Bij{ }

Matrices A and B have these dimensions:

[r x c] and [s x d]

Matrices A and B can be multiplied if:

[r x c] and [s x d]

c = s

The resulting matrix will have the dimensions:

[r x c] and [s x d]

r x d

A = a11 a12

a21 a22

B = b11 b12 b13

b21 b22 b23

++++++

=232213212222122121221121

2312131122121211 21121111

babababababa

babababababaC

[2 x 2]

[2 x 3]

[2 x 3]

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

[3 x 2] [2 x 3]A and B can be multiplied

=

=+=+=+=+=+=+=+=+=+

=1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

[3 x 3]

=

=+=+=+=+=+=+=+=+=+

=1 1 1

3 1 2

8 2 5

12*01*1 10*01*1 11*01*1

32*11*1 10*11*1 21*11*1

82*31*2 20*31*2 51*31*2

C

A =

2 3

1 1

1 0

and B =

1 1 1

1 0 2

[3 x 2] [2 x 3]

[3 x 3]

Result is 3 x 3

Inversi on

B−1B = BB−1 = I

Like a reciprocal in scalar math

Like the number one in scalar math

1st Precinct : x1 + x2 = 6

2nd Precinct : 2x1 + x2 = 9

First precinct: 6 arrests last week equally divided between felonies and misdemeanors.

Second precinct: 9 arrests - there were twice as many felonies as the first precinct.

=

9

6 *

1 2

1 1

2

1

x

x

=

3

3

2

1

x

x

1 2

1 1 Note: Inverse of is

−

−1 2

1 1

9

6*

1 2

1 1 *

1 2

1 1*

1 2

1 1

2

1

−

−=

−

−x

x Premultiply both sides by inverse matrix

3

3 *

1 0

0 1

2

1

=

x

x A square matrix multiplied by its inverse results in the identity matrix.

A 2x2 identity matrix multiplied by the 2x1 matrix results in the original 2x1 matrix.

aijxj = bi or Ax = bj=1

n

∑

x = A−1Ax = A−1b

n equations in n variables:

unknown values of x can be found using the inverse of matrix A such that

Select the cells in which the answer will appear

Follow the same procedure Select cells in which answer is to be displayed Enter the formula: =minverse( Select the cells containing the matrix to be

inverted Close parenthesis – type “)” Press three keys: Control, shift, enter

1) Enter “=mmult(“

2) Select the cells of the first matrix

3) Enter comma “,”

4) Select the cells of the second matrix

5) Enter “)”

Enter these three key strokes at the same time:

control

shift

enter

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