Algebraic Properties of Matrix Operations The m x n matrix with all entries of zero is denoted by 푶_풎풏 , for a matrix A of size m x n, we have:

1. HELLO!
ARE YOU READY? 
IF YES, LET’S
START.

2. LINEAR ALGEBRA
Solving a Matrix Equation
a) 3X + A = B
A =
Using your skills in matrix addition and scalar
multiplication, solve the following matrix equation:
𝟏 −𝟐
𝟎 𝟑
B =
−𝟑 𝟒
𝟐 𝟏
b) 2X + 2A = 4B

3. M117
Linear Algebra
Lesson 4
Algebraic Properties of Matrix
Operations
REPORTERS:
Joycee Anne Pura
Kristopher Hiloma
BSED ─ MATHEMATICS

4. LINEAR ALGEBRA
Let A, B and C be m x n matrices.
Properties of Matrix Addition
 A +B = B +A
 A + (B +C ) = (A +B ) + C
A =
𝟒 ￚ𝟏
𝟐 𝟑 B =
𝟏 ￚ𝟐
𝟓 𝟕
C = ￚ𝟓 𝟑
ￚ𝟕 ￚ𝟏𝟎
Commutativity of Addition
Associativity of Addition
A +B or B +A =
𝟓 ￚ𝟑
𝟕 𝟏𝟎
B +C =
ￚ𝟒 𝟏
ￚ𝟐 ￚ𝟑
𝟎 𝟎
𝟎 𝟎
Zero Matrix

5. LINEAR ALGEBRA
Let A and B be m x n matrices and c and d are scalars.
Properties of Scalar Multiplication
 (cd) A = c (dA)
 1A = A
 c (A + B ) = cA + cB
 (c + d) A = cA + dA
Associative Property of Multiplication
Multiplicative Identity
Distributive Property
Distributive Property
A =
𝟒 ￚ𝟏
𝟐 𝟑
B = 𝟏 ￚ𝟐
𝟓 𝟕
c = 2 d =
𝟏
𝟐

6. LINEAR ALGEBRA
Properties of Zero Matrices
 A + 𝑶 𝒎𝒏 = A
A =
𝟒 ￚ𝟐 ￚ𝟓
𝟐 𝟑 ￚ𝟏
𝑶 𝟐𝟑 =
𝟎 𝟎 𝟎
𝟎 𝟎 𝟎
The m x n matrix with all entries of zero is denoted by
𝑶 𝒎𝒏 , for a matrix A of size m x n, we have:
This property states that
𝑶 𝒎𝒏 or the zero matrix is
the additive identity for the
set of all m x n matrices.

7. LINEAR ALGEBRA
Properties of Zero Matrices
 A + D = 𝑶 𝒎𝒏
A =
This property states that ￚ A
or the negative of A is the
additive inverse of A .
A + (ￚ A ) = 𝑶 𝒎𝒏
𝟒 ￚ𝟐 ￚ𝟓
𝟐 𝟑 ￚ𝟏
ￚ A =
ￚ𝟒 𝟐 𝟓
ￚ𝟐 ￚ𝟑 𝟏
ￚ A = ￚ1(A)
The m x n matrix with all entries of zero is denoted by
𝑶 𝒎𝒏 , for a matrix A of size m x n, we have:

8. LINEAR ALGEBRA
Properties of Zero Matrices
 If cA = 𝑶 𝒎𝒏 , then c = 0 or A = 𝑶 𝒎𝒏 .
A =
𝟒 ￚ𝟐 ￚ𝟓
𝟐 𝟑 ￚ𝟏
cA =
A = 𝑶 𝒎𝒏
0
𝟒 ￚ𝟐 ￚ𝟓
𝟐 𝟑 ￚ𝟏
cA =
𝟎 𝟎 𝟎
𝟎 𝟎 𝟎
c = 𝟏𝟎
cA = 10
𝟎 𝟎 𝟎
𝟎 𝟎 𝟎
cA =
𝟎 𝟎 𝟎
𝟎 𝟎 𝟎
c = 0
The m x n matrix with all entries of zero is denoted by
𝑶 𝒎𝒏 , for a matrix A of size m x n and a scalar c, we
have:

9. LINEAR ALGEBRA
What did you notice?
For these first three properties of matrix
operations, “matrices behave like real numbers.”
Properties of Matrix
Addition
Properties of Zero
Matrices
Commutativity of Addition
Associativity of Addition
Additive Identity
Additive Inverse
Zero (0) as a Multiplier
Properties of Scalar Multiplication

10. LINEAR ALGEBRA
Let A, B and C be matrices with sizes such that the
given matrix products are defined and c be a scalar.
Properties of Matrix Multiplication
a) A (BC) = (AB) C
b) (A + B) C = AC + BC
c) C (A + B) = CA + CB
d) c (AB) = (cA)B + A(cB)
Associativity Matrix Product
Right Distributive Property
Left Distributive Property

11. LINEAR ALGEBRA
Properties of Identity Matrices
𝑰 𝟏 =
For a positive integer, 𝑰 𝒏 would denote the square matrix of
order n whose main diagonal (left to right) entries are 1 and
the rest of the entries are zero (Identity Matrix).
𝟏 𝟎
𝟎 𝟏
𝑰 𝟑 =
𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
[𝟏] 𝑰 𝟐 =

12. LINEAR ALGEBRA
Properties of Identity Matrices
a) A 𝑰 𝒏 = A
A =
𝑰 𝒏 =
b) 𝑰 𝒎A = A
A =
For a positive integer, 𝑰 𝒏 and 𝑰 𝒎 would denote the square matrices
of order n and m, respectively, whose main diagonal (left to right)
entries are 1 and the rest of the entries are zero (Identity Matrix).
𝟑 −𝟐
𝟒 𝟎
−𝟏 𝟏
𝟏 𝟎
𝟎 𝟏
𝑰 𝒎 =
𝟏 𝟎 𝟎
𝟎 𝟏 𝟎
𝟎 𝟎 𝟏
−𝟐
𝟏
𝟒
Multiplicative Identity

13. LINEAR ALGEBRA
Let A, B and C be matrices with sizes such that the
given matrix products are defined and c be a scalar.
Properties of Transposes
a) (𝑨 𝑻
)
𝑻
= A
b) (A + B )
𝑻
= 𝑨 𝑻
+ 𝑩 𝑻
c) (cA)
𝑻
= c(𝑨 𝑻
)
d) (AB )
𝑻
= 𝑩 𝑻
𝑨 𝑻
Transpose of a transpose
Transpose of a sum
Transpose of a scalar multiple
Transpose of a product

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