Introduction to Matrices (lecture)
A matrix is an array of numbers, called elements, which are set in
rows and columns.
• The order/dimensions of a matrix is (rows x columns)
o Row matrix: a matrix consisting of a single row.
o Column matrix: a matrix consisting of a single column.
o Square matrix: a matrix with, rows = columns
o Zero matrix (null matrix): a matrix with all 0’s.
Matrix Addition
• Only matrices of the same order can be added
o Addition is associative: 𝐴 + (𝐵 + 𝐶 ) = (𝐴 + 𝐵) + 𝐶
o Addition is commutative: 𝐴 + 𝐵 = 𝐵 + 𝐴
Scalar Multiplication
A scalar is a number that has magnitude but no direction.
• In scalar multiplication, the scalar is multiplied with all the
elements of the matrix.
• 𝐴 = 2𝐵 ⟺
1
2
𝐴=𝐵
• If A and B have the same order, then for any scalar h and k:
o ℎ(𝐴 + 𝐵) = ℎ𝐴 + ℎ𝐵
o (ℎ + 𝑘)𝐴 = ℎ𝐴 + 𝑘𝐴
o ℎ(𝑘𝐴) = (ℎ𝑘)𝐴
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Q1. Given 𝐴 = (
1 −2
4 0
) , 𝐵=(
) , calculate,
2 3
2 −1
a. 𝐴 + 3𝐵
b. 𝐵 − 4𝐴
Matrix Multiplication
• If matrix A is multiplied to matrix B to give AB, then,
o the dimensions of A and B should be of the form
(𝑝 × 𝑞) × (𝑞 × 𝑟) ,
Number of columns of A = Number of rows of B.
o The product matrix will be of the order (𝑝 × 𝑟).
• The product 𝐴𝐵 means, 𝐴 is post-multiplied by 𝐵 or that 𝐵 is
pre-multiplied by 𝐴.
• Matrix multiplication,
o Is generally, NOT commutative: 𝐴𝐵 ≠ 𝐵𝐴.
o Is Associative: (𝐴𝐵)𝐶 ≡ 𝐴(𝐵𝐶).
o Is Distributive over addition: (𝐴 + 𝐵)𝐶 ≡ 𝐴𝐶 + 𝐵𝐶.
1 −2
2
4 0
) , 𝐵=(
) , 𝐶=(
2 3
0.5
2 −1
−1 4 1
𝐷=( 1
0 2)
−3 −2 3
Q2. Given 𝐴 = (
−1 1
),
3 0
If possible, calculate
a. 𝐴𝐵
b. 𝐶𝐷
c. 𝐷2
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Identity Matrix
• The identity matrix (unit matrix), I, is a square matrix in
which the elements in the main diagonal (leading diagonal)
are all 1’s and all the other elements are 0’s.
• The main diagonal of a square matrix is the diagonal from
the top left element to the bottom right.
• Any matrix pre- or post-multiplied by I does not change:
𝐴𝐼 = 𝐼𝐴 = 𝐴
WORKSHEET 1
Determinant of a Matrix
• A scalar value associated with a square matrix that implies
certain characteristics of a linear transformation described
by the matrix is called a determinant.
• A singular matrix is the one with determinant = 0.
1. Det of a 2 x 2 Matrix
𝑎
If 𝑀 = (
𝑐
𝑏
) , then
𝑑
𝑎 𝑏
𝑑𝑒𝑡(𝑀) = |𝑀| = |
| = 𝑎𝑑 − 𝑐𝑏.
𝑐 𝑑
2 −5
Q2. Find the determinant of the matrix (
).
6 7
𝑎−3
−2
Q3. Let 𝑀 = (
), where 𝑎 is a constant. Find the set of
2
𝑎+2
possible values of 𝑎 for which 𝑀 is singular.
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2. Det of a 3 x 3 Matrix
𝑎
If 𝑀 = (𝑑
𝑔
𝑏
𝑒
ℎ
𝑐
𝑓 ) , then
𝑖
𝑒
det(𝑀) = 𝑎 |
ℎ
𝑑
𝑓
|−𝑏|
𝑔
𝑖
𝑓
𝑑
|+𝑐|
𝑔
𝑖
𝑒
|
ℎ
𝑒 𝑓
• 𝐴=|
| is called the minor of the element 𝑎,
ℎ 𝑖
𝑑 𝑓
𝐵=|
| is the minor of the element 𝑏 and
𝑔 𝑖
𝑑
𝐶=|
𝑔
𝑒
| is the minor of the element 𝑐.
ℎ
2
Q4. Find the determinant of the matrix (0
1
4 5
1 2).
−1 3
• The determinant of a square matrix has the following
properties:
o If two rows (or two columns) are identical, then
det(𝑀) = 0 .
o If two rows (or two columns) are interchanged, then
only the sign of 𝑑𝑒𝑡(𝑀) is changed.
o A multiple of one row (or column) added to another
row (or column), does not change the determinant.
o If A, B are square matrices, then
𝑑𝑒𝑡(𝐴𝐵) = 𝑑𝑒𝑡(𝐴) × 𝑑𝑒𝑡(𝐵) .
o If M is an 𝑛 × 𝑛 matrix, then 𝑑𝑒𝑡(𝑘𝑀) = 𝑘 𝑛 𝑑𝑒𝑡(𝑀) .
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Elementary Row Operations
• Elementary row operations in Gaussian Elimination (Row
Reduction) method:
o Interchange any two rows.
o Multiply any row by a non-zero scalar.
o Add to any row, element by element, a scalar multiple of
the corresponding elements of another row.
• A matrix is said to be in row echelon form if the pivot
elements of each row have all zeros on their left and below.
• The determinant is the product of the elements of the
𝑎
leading diagonal in row echelon form, (
0
𝑎
𝑏
) , (0
𝑐
0
𝑏
𝑑
0
𝑐
𝑒 ),
𝑓
divided by the row multiples on the changing rows.
Q5.
Calculate the determinants of the following,
1
a. (1
2
4
b. (2
1
3
c. (0
1
1 0
3 1)
1 4
1 −1
4
0)
−1 5
−1 −3
4
1)
−1 2
WORKSHEET 2
Matrix Inverse
• Only non-singular matrices (determinant ≠ 0) have an inverse.
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1. Inverse of a 2 x 2 matrix
𝑎
If 𝑀 = (
𝑐
𝑏
),
𝑑
then 𝑀−1 =
1
𝑑
(
det (𝑀) −𝑐
−𝑏
) , 𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑑 det (𝑀) ≠ 0
𝑎
• 𝑀𝑀−1 = 𝑀−1 𝑀 = 𝐼
Q6. Given that B and C are non-singular matrices and 𝐴𝐵𝐶 = 𝐼, show that
𝐴 = 𝐶 −1 𝐵−1 .
Q7. Find the matrix M, given that 𝑀𝑁 = (
1 1
3 −1
) and 𝑁 = (
)
−1 1
−2 1
2. Inverse of a 3 x 3 matrix
• Use row reduction method on (𝐴|𝐼) to make (𝐼|𝐴−1 ).
Q8. Using row operations, find the inverse matrix of,
4
a. (2
3
0
b. (1
1
1 −1
0
1)
−2 4
0 1
2 0)
3 1
• (𝐴𝐵)−1 = 𝐵−1 𝐴−1 .
1 0
Q9. Given that 𝐴 = (0 1
1 1
1
−1
)
𝐵
=
(
and
1
1
0
0
1 0
0 1 ), determine the
0 −1
result of 𝐶, where 𝐴𝐵𝐶 = 𝐼.
WORKSHEET 3
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