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106 LearnersLast updated on October 24, 2025
Properties of Matrices
Matrices have special properties that simplify operations. These properties help in adding, subtracting, multiplying, and finding the inverse of matrices. We'll also look at identity matrices, which act like the number 1 in multiplication, and inverse matrices, which reverse the effect of another matrix.
What are the Properties of Matrices?
Matrices have properties that help us compute inverses and perform operations like addition, subtraction, and multiplication. Here are 5 properties mentioned below.
1. Properties of Matrix Addition
2. Properties of Scalar Multiplication of a Matrix
3. Properties of Matrix Multiplication
4. Properties of the Transpose of a Matrix
5. Properties of Inverse Matrices.
Properties of Matrix Addition
1. Commutative Law
For any two matrices A and B of the same size (m×n), adding them works either way:
A + B = B + A because addition happens entry-wise, as each pair of corresponding elements is added. Since addition is commutative (aij+bij = bij+aij), the order doesn’t matter.
2. Associative Law
For any three matrices A, B, C of the same size:
A + (B + C) = (A + B) + C — how you group the addition doesn’t matter.
3. Additive Identity
There exists a special zero matrix O (all entries are 0) such that for any matrix A of the same size:
A + 0 = 0 + A = A.
4. Additive Inverse
Every matrix A has a negative version, –A, (obtained by changing every entry’s sign). If you add them, you get zero:
A + (–A) = (–A) + A = 0.
Properties of Scalar Multiplication of a Matrix
Scalar multiplication scales a matrix by multiplying each entry by a real number. Its properties include compatibility with scalar operations.
The product of a constant and a sum of matrices means multiplying each matrix by the same number and adding them.
- k(A + B) = kA + kB
- Similarly, adding two constants before multiplying gives the same result as adding their separate products:
(k + l)A = kA + lA
A and B are the same size matrices, and k, l are scalars.
Properties of the Transpose of a Matrix
When A and B are the same size (m×n) and k is any number, the transpose has these key rules: Transposing twice returns the original. Transposition distributes over addition and scalars (A+B)T = AT + BT, (kA)T = kAT. It also reverses multiplication order (AB)T = BTAT.
- Transposing a matrix twice brings you back to the original, so (A′)′ = A.
- When you transpose a matrix that's been multiplied by a number, the number stays outside: (kA)′ = kA.
- showing that transposition can be applied term‑by‑term when adding matrices. (A+B)′ = A′+B′.
- Finally, when you transpose a product of two matrices, the order flips: (AB)′ = B′A′.
Properties of a Square Matrix
A square matrix has the same number of rows and columns. Its trace and the sum of its diagonal entries provides insight into transformations. Square matrices have a determinant, which indicates invertibility if non-zero. The transpose of a square matrix simply flips rows and columns. The identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere.
Common Mistakes of Properties of Matrices and How to Avoid Them
Students often make mistakes like mismatching dimensions, losing negative signs, misapplying rules, or forgetting that matrix multiplication isn’t commutative. Recognizing these errors early ensures correct answers.
Mistake 1
Mixing up matrix dimensions
Adding or multiplying matrices of incompatible sizes may lead to errors. Always check dimensions—rows × columns must match.
For example, trying to add a 2×3 matrix to a 3×2 matrix fails.
Mistake 2
Wrong transpose of a product
Writing (AB)′=A′B′ is wrong; always use the correct one, which is (AB)′=B′A′.
For example, \(A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix}\) \(B = \begin{pmatrix} 0 & 1 \ 0 & 1 \end{pmatrix}\)
then
(AB)′ = B′A′ ≠ A′B′.
Mistake 3
Incomplete additive inverse
Forgetting to flip negative entries. We need to be more focused while changing every entry’s sign, including negatives.
For example, transforming [−2, 3] into [2, 3] instead of [2,−3].
Mistake 4
Dropping a scalar in the transpose
Ignoring scalars when transposing. Keep the scalar outside the transpose: (kA)′ = kA′.
For example, miswriting (kA)T = kAT.
Mistake 5
Swapping rows instead of transposing rows to columns
Exchanging rows instead of swapping them with columns. Ensure the element at row i, column j moves to row j, column i.
For example, row 1 swapped with row 2 instead of column 1 swapped with row 2.
Real-Life Applications of Properties of Matrices
Matrices help us solve real-life problems, such as recommend shows on Netflix, predict weather with numerical models, edit images, plan routes for GPS navigation, and analyze social media networks.
- Computer Graphics & Animation: Matrices handle scaling, rotation, and moving objects smoothly by chaining transformations.
- Solving Linear Equations: Instead of solving each equation one by one, you can solve many at once using matrix inverses or elimination.
- Image & Signal Processing: Pictures are matrices of pixels. Operations like sharpening, smoothing, and filtering are done via matrix multiplication.
- Google’s PageRank: Web pages link in complex ways. A matrix models these links and ranks pages by analyzing that structure.
- Analyzing social networks: Think of people as dots, connections as lines, then make a matrix. You can multiply and add matrices to find groups or influencers. Think of people as dots, connections as lines, then make a matrix. You can multiply and add matrices to find groups or influencers.
Solved Examples of the Properties of Matrices
Problem 1
If (A+B)2 = A2 + 2AB + B2, what can you conclude about matrices A and B?
A and B commute, which is AB = BA.
Explanation
Combine by summing coefficients of matching terms:
For (A+B)2 = A2+2AB+B2 = A2 + 2AB + B2, expanding gives A2+AB+BA+B2. Equality holds only if AB=BA.
Problem 2
Prove (A′)′ = A using a specific matrix.
For example, A = . It's transpose A′ = , and transposing again returns the original A.
Explanation
Transposing twice reverses row–column swaps, restoring the original matrix.
For example, if A = then (A′)′ = A.
Problem 3
Under what condition does Under what condition does A2 − B2 = (A−B)(A+B) hold?A2 − B2 = (A−B)(A+B) hold?
When AB = BA.
Explanation
Expanding (A−B)(A+B) yields A2+AB−BA−B2. Equality to A2−B2 demands AB−BA = 0, meaning A and B commute.
Problem 4
When A and B are idempotent and AB = BA = 0, is A + B idempotent?
Yes—(A+B)2 = A+B.
Explanation
Since A2 = A, B2 = B, and AB = BA = 0, (A+B)2 = A2 + AB + BA + B² = A + B, so A + B is idempotent.
Problem 5
What is the transpose of a matrix?
The transpose of a matrix A, denoted AT, is obtained by swapping its rows and columns.
Explanation
If A = [aij], then AT = [aji]. Transposing a matrix twice returns the original matrix (AT)T=A.
FAQs of the Properties of Matrices
1.What is a Matrix?
2.What are the Properties of a Matrix?
3.What are the Properties of a 3×3 Matrix?
4.What is the Formula for a Matrix?
5. What is the Zero‑matrix Rule?
Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
