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Matrices
- A matrix is an array of numbers, each number being called an element.
- Matricies are denoted by capital letters.
- An r c ( r by c ) has r rows and c columns.
- r c is the order of the matrix.
A
^
has the order 4 3
Equal Matrices
Same order and identical elements.
Square Matrices
Same number of rows & columns. i.e. 2 2 , 3 3
Addition and Subtraction
Add or subtract the corresponding elements. The matrices must be of
the same order.
^
- Addition is commutative (independent of order)
A B B A
- Addition is associative (independent of the grouping of elements)
A^ ^ B^ ^ C^ ^ A^ ^ B^ C
Multiplication by a constant
Multiply each element by the constant.
e .g.
e .g.
e .g.
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Basic Matrix Operations Date________________ Period____
Simplify. Write "undefined" for expressions that are undefined.
3 6 −1 − −5 −
0 − 6 0 2 3
−5 2 − 4 −2 0 − 6 −5 − 1 3 −
- − 5 6 − 4 −2 −
- − −3 0 0 5
- 4 2 + −2 −
- 5 4 3
- −5 1 −2 −1 2
- 5 5 1 1 − 1 2
- −2 u 7 u 3 w
5 u 5
2 4
5 6
- 4 − 3 −
−4 n n + m −2 n −4 n
4 − 3 m 0
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Basic Matrix Operations Date________________ Period____
Simplify. Write "undefined" for expressions that are undefined.
3 6 −1 − −5 −
0 − 6 0 2 3 3 5 5 − −3 2
−5 2 − 4 −2 0 − 6 −5 − 1 3 − −11 7 4 3 −5 3
- − 5 6 − 4 −2 − −25 −30 20 −20 10 5
- − −3 0 0 5 15 0 0 −
- 4 2 + −2 − 2 −
- 5 4 3 20 15
- −5 1 −2 −1 2 −5 10 5 −
- 5 5 1 1 − 1 2 25 5 5 − 5 10
- −2 u 7 u 3 w
5 u 5 −14 u
−6 uw
−10 u
−10 u
2 4
5 6 7 10
- 4 − 3 − − 12 −
−4 n n + m −2 n −4 n
4 − 3 m 0 −4 n + 4 n + m − 5 −2 n + 3 m −4 n
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13) 2 −5 −3 + 1 −2 −
3 −7 −
x + y x − 6
− −6 xy x + y − 5 x − 6 − 6 xy
- 4 c 0 6 0 3 a 0 24 c 0 12 ca
- −3 y −2 x − y −4 y −3 x 6 yx 3 y
12 y
9 yx
- 3 2 u v
u
6 u 3 v
3 u
- − x − 1 −2 x −5 y − y −2 −3 x − x − 1 − y −2 x + 2 −5 y + 3 x
−6 r + t − r 6 s
6 r −4 t −3 r + 2 t − r − 4 t 6 s − 3 r + 2
z − 5 − −1 − 6 z 3 y
−3 y 3 z 5 + z 4 z z − 5 − 3 y −6 + 3 z 4 − 5 z 3 y + 4 z
- 5 6 1 2 −6 − 1 6 −6 6 29 −1 16 −
−10 10 −
5 3 5 1 − −6 0 1 − − 5 4 −2 − 6 − 6 11
1 1 6 − 0 0
- 5 −4 6 1 1 −4 − −19 31 11 1 −20 − -2- Create your own worksheets like this one with Infinite Algebra 2. Free trial available at KutaSoftware.com
Matrix Multiplication
For matrix multiplication, M N is only possible if,
“ N u m b e r o f c o lu m n s o f M N u m b e r o f r o w s o f N ”
a c
M
b d
p r
N
q s
a p c q a r c s
M N
b p d q b r d s
^
^
M
N
M N
^ ^ ^
^ ^ ^
N M
^ ^
^
^
- Matrix multiplication is not commutative
M N N M
- Matrix multiplication is associative
L M^ N^ L^ M N
- If M is r c & N is c s then MN is r s
e.g. M is 4 3 , N is 3 2.
MN is 4 2
NM doesn’t exist!!
e .g.
2 3 3 2
2 2
3 3
3 2 2 ^3
s a m e o rd e r
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Matrix Multiplication Date________________ Period____
Simplify. Write "undefined" for expressions that are undefined.
0 2 −2 − ⋅ 6 − 3 0
6 − ⋅ −5 4
−5 − −1 2 ⋅ −2 − 3 5
−3 5 −2 1 ⋅ 6 − 1 −
0 5 −3 1 −5 1 ⋅ −4 4 −2 −
5 3 5 1 5 0 ⋅ −4 2 −3 4 3 −
− 6 0 ⋅ 3 −
3 2 5 2 3 1 ⋅ 4 5 − 5 −1 6
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Matrix Multiplication Date________________ Period____
Simplify. Write "undefined" for expressions that are undefined.
0 2 −2 − ⋅ 6 − 3 0 6 0 −27 12
6 − ⋅ −5 4 −30 24 15 −
−5 − −1 2 ⋅ −2 − 3 5 −5 − 8 13
−3 5 −2 1 ⋅ 6 − 1 − −13 − −11 −
0 5 −3 1 −5 1 ⋅ −4 4 −2 − −10 − 10 − 18 −
5 3 5 1 5 0 ⋅ −4 2 −3 4 3 − −14 − −19 22
− 6 0 ⋅ 3 − −15 5 18 − 0 0
3 2 5 2 3 1 ⋅ 4 5 − 5 −1 6 Undefined
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3 − −3 6 −6 − ⋅ −1 6 5 4 −8 14 33 6 −24 −
5 4 2 − ⋅ − 3 − −
−1 1 − 5 2 − 6 −5 1 −5 6 0 ⋅ 6 5 5 − 6 0 −7 − 10 13 17 60 0 −
−2 − −4 3 5 0 4 − ⋅ 2 −2 2 −2 0 − 8 4 14 −14 8 − 10 −10 10 20 −8 26
- 2 −5 v ⋅ −5 u − v 0 6 −10 u −32 v
−4 − y −2 x − ⋅ −4 x 0 2 y − 16 x − 2 y
5 y 8 x
− 8 y 20 Critical thinking questions:
- Write an example of a matrix multiplication that is undefined. Many answers. Ex: 1 2 3 4 ⋅ 1 2 3 4 5 6
- In the expression A ⋅ B , if A is a 3 × 5 matrix then what could be the dimensions of B? 5 × Anything
Create your own worksheets like this one with (^) Infinite Algebra 2. Free trial available at KutaSoftware.com
Identity Matrix
Denoted by I , such that
M I IM M
I (^) 1 1 , 2
I
, 3
I
1 0 0
0 1 0
0 0 1
n I
^
e.g.
M
M I
Determinant of a 2 by 2 Matrix
Example
Find
(a)
3 2 d e t 1 3
(b)
3 7
2 4
Question
Given that
3 2
x 6
is a singular matrix, find x
Example
3 2 2 1 , 2 5 4 0
A B
(^) ^ ^
. Findd e t (^) A B
Question
Given that
1 4
1 2
A
and
7 1 9
5 5
A B
, findd e t B
The determinant of the matrix
a b
c d
is
defined as a d b c
If a matrix has a determinant of zero,
then it is called a singular matrix
d e t (^) A B (^) d e t (^) A (^) d e t B
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Determinants of 2×2 Matrices Date________________ Period____
Evaluate the determinant of each matrix.
0 − −6 − −
−6 0 6 − 36
−1 1 −1 4 −
0 4 6 5 −
0 − 6 − 6
5 3 6 6 12 Evaluate each determinant.
−5 3 4 2 −
−9 − −7 − 27
−1 8 5 0 −
8 − −10 9 12
0 6 −8 0 48
10 − −7 3 −
−5 0 2 10 −
2 − 7 − 0
- Evaluate: 1 2 3 4
5 2 −2 6 32
- Give an example of a 2×2 matrix whose determinant is 13. Many answers. Ex: 4 13 1 5 Create your own worksheets like this one with (^) Infinite Algebra 2. Free trial available at KutaSoftware.com
Inverse Matrix
If M & N are square matrices and M N I then
N M
.
M
is the inverse matrix.
M M I
a c
M
b d
w y
M
x z
^
a c w y
b d x z
Find w , x , y , z in terms of a , b , c , d.
a w c x
a y c z
b w d x
b y d z
b a b w b c x b
a a b w c d x
d a d w c d x d
c b c w c d x
b c x c d x b
b
b c a d
b
x
a d b c
d
w
a d b
a d b c w d
c
w
d a d y c d y
c b c y c d z c
b a b y b c z
a a b y a d z a
b c y c d y c
c
b c a d
c
y
a d b c
a
z
a d b
a d b c z a
c
z
d c
a d b c a d b c
M
b a
a d b c a d b c
When
a c
M
b d
1 1 d c M a d b c b^ a
- (^) a d b c is known as the determinant of the matrix M , denoted by
a c
b d
- If (^) a d b c 0 , the inverse matrix doesn’t exist. M is said to be singular
- If a d b c 0 , the inverse exists. M is said to be non-singular
Inverse of a 2 by 2 matrix
Example
Find the inverse of
2 4
1 3
Question
2
3 4
3
A k
(a) Find the two possible values for which A has no inverse
(b) Show that when k 2 , A is self-inverse (i.e. the inverse is the same as the original
matrix)
Example
9 2 3 4 , 3 1 7 6
A B
Find
(a)
1 A
(b)
1 B
(c)
1 B A
The identity matrix is
1 0
0 1
I
The inverse
1 A
of the matrix A has the property 1 1 A A A A I
The inverse of the matrix
a b
c d
is given by
1
d e t
d b
A c^ a
If a matrix is singular then it has no inverse
(^1 1 ) A B B A
(^)
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Inverse Matrices Date________________ Period____
For each matrix state if an inverse exists.
−9 − −2 −
−2 1 −6 1
4 − −9 6
0 0 −6 4 Find the inverse of each matrix.
11 − 2 −
0 − −1 −
−1 7 −1 7
1 − −6 −
