Matrices and Determinants
Matrices
 A matrix is a rectangular arrangement
of numbers in rows and columns. Rows
run horizontally and columns run
vertically.
 The dimensions of a matrix are stated
“m x n” where ‘m’ is the number of
rows and ‘n’ is the number of columns.
Equal Matrices
Two matrices are considered equal
if they have the same number of
rows and columns (the same
dimensions)
AND
all
their
corresponding elements are exactly
the same.
Types of Matrices
1. Rectangular Matrix
7. Row Matrix
2. Square Matrix
8. Column Matrix
3. Diagonal Matrix
9. Upper
4. Scalar Matrix
Matrix
10. Lower
Triangular
Matrix
11. Sub matrix.
5. Identity Matrix
6. Null Matrix
Triangular
Matrix Addition
 You can add or subtract matrices if
they have the same dimensions (same
number of rows and columns).
 To do this, you add (or subtract) the
corresponding numbers (numbers in
the same positions).
Matrix Addition
Example:
 2 4  1 0 
5 0    2 1  

 

1 3  3 3
 3 4 
 7 1


 2 0 
Properties of Matrix Addition
Matrix addition is commutative i.e.
A+B = B+A
Matrix addition is associative i.e.
(A+B)+C = A+(B+C)
Matrix addition is distributive w.r.t. scalar K
K(A+B) = KA+KB
Scalar Multiplication
 To do this, multiply each entry in
the matrix by the number outside
(called the scalar). This is like
distributing a number to a
polynomial.
Scalar Multiplication
Example:
 2 4   8 16 




4 5 0   20 0


1 3  4 12 
Matrix Multiplication


Matrix
Multiplication
is
NOT
Commutative! Order matters!
You can multiply matrices only if the
number of columns in the first matrix
equals the number of rows in the second
matrix.
2 columns
3
2
 5 6    1

  3
 9 7 
2 0 
4 5 
2 rows
Matrix Multiplication

Take the numbers in the first row of
matrix #1. Multiply each number by its
corresponding number in the first
column of matrix #2.
Total these
products.
3
2
 5 6    1

  3
 9 7 
2 0 
4 5 
2 1  3 3  11
The result, 11, goes in row 1,
column 1 of the answer.
Repeat with row 1, column 2;
row 1 column 3; row 2, column
1; ...
Matrix Multiplication

Notice the dimensions of the matrices and
their product.
3
2
 5 6    1

  3
 9 7 
3x2
__
8 15 
 11
2 0  


13
34
30

4 5  
 12 46 35 
2 x 3__
3 x 3__
__
Matrix Multiplication

Another example:
2 1
 9 0    5  

  2 
10 5  
3x2
2x1
 8 
 45


 60 
3x1
Properties of Matrix
Multiplication
Matrix Multiplication is not commutative,
i.e. AB ≠ BA
Matrix Multiplication is associative, i.e.
A(BC) = (AB)C
Matrix Multiplication is distributive, i.e.
A(B+C) = AB+AC
Special Types of Matrices
Idempotent Matrix
Nilpotent Matrix
Involutory Matrix
Transpose of Matrix
Let A be any matrix. The matrix obtained
by interchanging rows and columns of A is
called the transpose of A and is denoted
by A’ or AT.
Properties of Transpose of
Matrices
1. The transpose of transposed matrix is
equal to the matrix itself, i.e. (A’)’ = A.
2. The transpose of the sum of the two
matrices is equal to the transpose of the
matrices, i.e. (A+B)’ = A’+B’.
3. The transpose of the product of two
matrices is equal to the product of their
transposes in the reverse order, i.e.
(AB)’ = B’A’.
Matrix Determinants
 A Determinant is a real number associated
with a matrix. Only SQUARE matrices
have a determinant.
 The symbol for a determinant can be the
phrase “det” in front of a matrix variable,
det(A);
or
vertical
bars
around
a matrix, |A| or 3 1 .
2
4
Determinant of a 2x2 matrix
1
3

 1 0   3  1
-½
0
2   3 2
Determinant of a 3x3 matrix
Imagine crossing out the first row.
And the first column.
-3
8
¼
2
0
-¾
4
180
11
Now take the double-crossed element. . .
And multiply it by the determinant of the
remaining 2x2 matrix


 3  0 11
  3
4 180 
Determinant of a 3x3 matrix
•Now take the negative of the doublecrossed element.
•And multiply it by the determinant of the
remaining 2x2 matrix.
•Add it to the previous result.
Now keep the first row crossed.
Cross out the second column.
-3
8
¼
2
0
-¾
4
180
11


 3  0 11   3
4 180  8  211    3 4   4
Determinant of a 3x3 matrix
Finally, cross out first
row and last column.
•Now take the double-crossed element.
•Multiply it by the determinant of the
remaining 2x2 matrix.
•Then add it to the previous piece.
-3
8
¼
2
0
-¾
4
180
11
 4  180  8  211    3 4   4

  1    2 180    0  4    695
4
 3  0 11   3
Computation
Method of Cofactors
Also known as the expansion of minors
Method of Minors
Determinant of a 2 x 2
matrix is difference in
products of diagonal
elements.
4 1 
A
 A  4  2  1 1  7

1 2 
General form for 2 x 2 matrix
a b 
A

c d 
A
ad - bc
Then,
What about larger matrices?
Use method of cofactors
Need to define a new term, “minor”
– Minor of an element aij is the determinant of
the matrix formed by deleting the ith row and
jth column
Example
1 2 3  a11
A  2 2 1   a21

 
3 1 4  a31
a12
a22
a23
a13 
a23 

a33 
Minor of a12 = 2 is determinant of the 2 x 2
matrix obtained by deleting the 1st row and 2nd
column
1 2 3  a11
A  2 2 1   a21

 
3 1 4  a31
a12
a22
a23
a13 
a23 

a33 
Minor of a12 is
2 1
3 4
83  5
Minor of a13 = 3 is
2 2
3 1
 2  6  4
Cofactors
Definition
– The cofactor of aij = (-1)i+j x minor
Evaluate cofactors for first three elements
of the 3 x 3 matrix
A11(-1)1+1
=1
A12(-1)1+2
A13(-1)1+3
=-1
=1
Pattern of signs
+ - +
- + + - +
Matrix of Cofactors
 c11 c12

C  c21 c22

c31 c32
c13 

c23

c33 
Determinant obtained by expanding
along any row or column of matrix of
cofactors
Determinant of A given by
A  a11c11  a12 c12  a13c13
Determinant of A
Element
a11 = 1
a12 = 2
a13 = 3
Minor
Cofactor
7
Element x
Cofactor
7
5
-5
-10
 4
-4
-12
2 1
7
1 4
2 1
3 4
2 2
3 1
Determinant of A = -15
Determinants of 4 x 4 matrices
Computational energy increases as order
of matrix increases
Use pivotal condensation (computer
algorithm)
Key Properties of Determinant
1. Determinant of matrix and its transpose
are equal.
2. If any two adjacent rows(columns) of a
determinant are interchanged, the value of
the determinant changes only in sign.
3. If any two rows or two columns of a
determinant are identical or are multiple of
each other, then the value of the
determinant is zero.
4. If all the elements of any row or column of
a determinant are zero, then the value of
the determinant is zero.
5. If all the elements of any row (or column) of a
determinant are multiplied by a quantity (K), the
value of the determinant is multiplied by the
same quantity.
6. If each element of a row (or column) of a
determinant is sum of two elements, the
determinant can be expressed as the sum of
two determinants of the same order.
7. The addition of a constant multiple of one row
(or column) to another row (or column) leave
the determinant unchanged.
8. The determinant of the product of two matrices
of the same order is equal to the product of
individual determinants.
Adjoint of a Matrix
If A is any square matrix, then the adjoint
of A is defined as the transpose of the
matrix obtained by replacing the element
of A by their corresponding co-factors.
Adj.A = Transpose of the cofactor matrix
Inverse Matrix
Inverse of square matrix A is a matrix A-1
that satisfies the following equation
– AA-1 = A-1A = I
Steps to success in Matrix
Inversion
If the determinant = 0, the inverse does
not exist if the matrix is singular.
Replace each element of matrix A, by it’s
minor
Create the matrix of cofactors
Transpose the matrix of cofactors
– Forms the adjoint
Divide each element of the adjoint by the
determinant of A.
Matrix Inversion
Pre multiplying both sides of the last equation by
A-1, and using the result that A-1A=I,
we can get
C'
1
1
1
 A , or A  adj A
A
A
This is one way to invert matrix A!!!
Matrix Inversion
Example
3 2 
A

1
0


 C11
C
 C21
A  2  0  inverse exists
C12   0 1

C22   2 3 
 0 2 
C '  adjA  

 1 3 
1
1  0 2   0 1 
A 
adj A   
 1


A
2  1 3   2  23 
1
Properties of Inverse Matrices
If A and B are non-singular matrices of the
same order, then (AB)-1 = B-1.A-1
The inverse of the transpose of a matrix is
equal to the transpose of the inverse of
that matrix, i.e. (A’)-1 = (A-1)’
The inverse of the inverse of a matrix is
the matrix itself i.e. (A-1)-1 = A
Cramer’s Rule
Given an equation system Ax=d where A is n x n.
1
x  A d  (adj A) d
A
1
1
x1 
A1
A
method of inverse
Cramer's Rule
|A1| is a new determinant were we replace the first column of |A|
by the column vector d but keep all the other columns intact
Cramer’s Rule
The expansion of the |A1| by its first column (the d column) will yield
the expression
n
d
i 1
i
Ci1
because the elements di now take the place of elements aij.
1
x1 
A1
A
Cramer’s Rule
In general,
a11
a12
d1
a1n
1 a21 a22
xj 

A
A
an1 an 2
d2
a2 n
dn
ann
Aj
This is the
statement of
Cramers’Rule
Cramer’s Rule
Find the solution of
5 x1  3x2  30
6 x1  2 x2  8
5 3
A
  28
6 2
A1 84
x1 

3
A 28
140
x1 

5
A
28
A2
30 3
A1 
  84
8 2
5 30
A2 
  140
6 8
Cramer’s Rule
Find the solution of the equation system:
7 x1  x2  x3  0
10 x1  2 x2  x3  8
♫ Work this out!!!!
6 x1  3x2  2 x3  7
A  61,
A1  61,
A2  183,
A3  244,
Cramer’s Rule
Solutions:
x1 
A1
A

61
1
61
183
x2 

3
A
61
A2
244
x3 

4
A
61
A2
Note that |A| ≠ 0 is necessary condition for the application of
Cramer’s Rule. Cramer’s rule is based upon the concept of the
inverse matrix, even though in practice it bypasses the process of
matrix inversion.
Rank of a Matrix
The number ‘r’ is called the rank of the
matrix A if
1. There exists at atleast one non-zero
minor of order r of A
2. Every minor of order (r+1) of A is zero.
The rank of a matrix A is denoted by p(A).
Steps to Find Rank of a Matrix
1. The given matrix should be a square matrix. If it
is not so, the matrix should be made a square
matrix by deleting the extra row or the column.
2. Find the determinant of the square matrix given
or obtained after deleting extra row or column.
3. If determinant of the matrix is zero, then take
the sub-matrix of the given matrix. Of the
determinant of any one of the sub-matrices is
not zero, then the order of that sub matrix
would be the rank of the given matrix.