Class 12 Maths · Original EZT practice
Matrices (Class 12 Chapter 3): summary and practice
A matrix is just a rectangular table of numbers - but the rules for adding and multiplying those tables power everything from computer graphics to solving equations. All questions are original EZT tutor practice following the chapter's own section order.
8 original worked examples, 8 fresh self-check questions and 4 mixed questions, with explained answers. These are not copied textbook exercises, an official chapter index or a full answer key. No affiliation with NCERT is claimed.
Official sources for topic and edition context, checked 21 October 2026:
1. Matrix and its order
A matrix is a rectangular array of numbers in rows and columns. Its order is rows x columns: a matrix with m rows and n columns is an m x n matrix.
2. Types of matrices
Row, column, square, diagonal, scalar, identity and zero matrices each have a defining shape. Two matrices are equal only when same order and every entry matches.
3. Operations on matrices
Add or subtract matrices entry by entry (same order only). Multiply by a scalar entry by entry. Matrix multiplication needs columns of the first equal to rows of the second, and in general AB is not BA.
4. Transpose of a matrix
The transpose A^T swaps rows and columns. Key properties: (A^T)^T = A, (kA)^T = kA^T, (A + B)^T = A^T + B^T and (AB)^T = B^T A^T.
5. Symmetric and skew symmetric matrices
A square matrix is symmetric when A^T = A and skew symmetric when A^T = -A (diagonal entries must be zero). Every square matrix splits into a symmetric plus a skew symmetric part.
6. Invertible matrices
If AB = BA = I for square matrices A and B, then B is the inverse of A, written A^-1. When an inverse exists, it is unique.
Choose, solve, check, then try a fresh question.
Before calculating, read the method prompt and name the condition that makes it useful. Write your own attempt. Open the steps only when you need them, and use the check to catch a wrong sign, unit or interpretation. Finish with the fresh question before reading its answer.
Read the order of a matrix
Choose the method: Count rows first, then columns.
EZT practice question
A matrix has 3 rows and 2 columns. What is its order, and how many entries does it have?
Show the worked solution
- Order is rows x columns = 3 x 2.
- Entries = 3 times 2 = 6.
Answer: Order 3 x 2, with 6 entries.
Check: A 3 x 2 matrix is tall, not wide - the first number is always rows.
Watch for this: Do not write 2 x 3; rows come first.
Now try it yourself
What is the order of a matrix with 4 columns and 1 row?
Check your answer
1 x 4.
One row, four columns.
Add matrices entry by entry
Choose the method: Same order required; add matching positions.
EZT practice question
A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]]. Find A + B.
Show the worked solution
- Add matching entries: 1 + 0, 2 + 1 in the first row.
- Then 3 + 1, 4 + 0 in the second row.
Answer: A + B = [[1, 3], [4, 4]].
Check: Each entry is the sum of the two entries in the same position.
Watch for this: Addition is only defined when both matrices have the same order.
Now try it yourself
Find B + A for the same matrices.
Check your answer
[[1, 3], [4, 4]] - the same.
Matrix addition is commutative: A + B = B + A.
Multiply two matrices
Choose the method: Row of the first times column of the second, entry by entry.
EZT practice question
A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]]. Find AB and BA. Are they equal?
Show the worked solution
- AB: first row [1 x 0 + 2 x 1, 1 x 1 + 2 x 0] = [2, 1]; second row [3 x 0 + 4 x 1, 3 x 1 + 4 x 0] = [4, 3].
- BA: first row [0 + 3, 0 + 4] = [3, 4]; second row [1 + 0, 2 + 0] = [1, 2].
Answer: AB = [[2, 1], [4, 3]] and BA = [[3, 4], [1, 2]]. No, AB is not equal to BA.
Check: Matrix multiplication is not commutative in general - this pair proves it.
Watch for this: Do not multiply entry by entry like addition; each entry is a row-column dot product.
Now try it yourself
Can a 3 x 2 matrix multiply a 2 x 4 matrix, and what is the product's order?
Check your answer
Yes, and the product is 3 x 4.
Columns of the first (2) match rows of the second (2); the outer numbers give the order.
Transpose a matrix
Choose the method: Rows become columns.
EZT practice question
C = [[1, 2, 3], [4, 5, 6]]. Write C^T and compute C times C^T.
Show the worked solution
- C^T = [[1, 4], [2, 5], [3, 6]] - each row turns into a column.
- C times C^T: entry one-one is 1 + 4 + 9 = 14; off-diagonal is 4 + 10 + 18 = 32; entry two-two is 16 + 25 + 36 = 77.
Answer: C^T = [[1, 4], [2, 5], [3, 6]] and C C^T = [[14, 32], [32, 77]].
Check: C C^T is always symmetric - here the off-diagonal entries match.
Watch for this: A 2 x 3 matrix transposes to 3 x 2, not another 2 x 3.
Now try it yourself
What is (C^T)^T?
Check your answer
C itself.
Transposing twice returns the original matrix.
Use the transpose of a product
Choose the method: (AB)^T = B^T A^T - the order flips.
EZT practice question
Why is (AB)^T = B^T A^T and not A^T B^T?
Show the worked solution
- In AB, rows of A meet columns of B.
- After transposing, rows of B^T must meet columns of A^T, so B^T comes first.
Answer: The product reverses: (AB)^T = B^T A^T.
Check: Try it on A = [[1, 2], [3, 4]], B = [[0, 1], [1, 0]]: (AB)^T = [[2, 4], [1, 3]] and B^T A^T gives the same.
Watch for this: Writing A^T B^T is the most common transpose error; the order always reverses.
Now try it yourself
What is (A + B)^T?
Check your answer
A^T + B^T.
Transpose distributes over addition without flipping.
Test symmetric and skew symmetric matrices
Choose the method: Compare A with A^T.
EZT practice question
Is P = [[1, 2], [2, 3]] symmetric? Is Q = [[0, 5], [-5, 0]] skew symmetric?
Show the worked solution
- P^T = [[1, 2], [2, 3]] = P, so P is symmetric.
- Q^T = [[0, -5], [5, 0]] = -Q, and the diagonal of Q is all zero, so Q is skew symmetric.
Answer: P is symmetric; Q is skew symmetric.
Check: A skew symmetric matrix must have zero diagonal, because a diagonal entry would need to equal its own negative.
Watch for this: Do not call a matrix skew symmetric if any diagonal entry is non-zero.
Now try it yourself
How do you build a symmetric matrix from any square matrix M?
Check your answer
Compute M + M^T.
(M + M^T)^T = M^T + M, which equals itself.
Verify an inverse matrix
Choose the method: Multiply both ways and look for I.
EZT practice question
A = [[1, 2], [3, 4]] and B = [[-2, 1], [3/2, -1/2]]. Show that B is A^-1.
Show the worked solution
- AB: row one [1 x -2 + 2 x 3/2, 1 x 1 + 2 x -1/2] = [1, 0]; row two [3 x -2 + 4 x 3/2, 3 x 1 + 4 x -1/2] = [0, 1].
- BA gives the same identity matrix.
Answer: AB = BA = I, so B = A^-1 and A is invertible.
Check: The inverse, when it exists, is unique - no second matrix can do this job.
Watch for this: Checking only AB is not enough in general; the definition needs AB = BA = I.
Now try it yourself
What is A times A^-1 for any invertible matrix A?
Check your answer
The identity matrix I.
That is the defining property of an inverse.
Understand what invertible means
Choose the method: There must exist a B with AB = BA = I.
EZT practice question
What does it mean for a square matrix A to be invertible?
Show the worked solution
- There must exist a square matrix B of the same order with AB = BA = I.
- Then B is written A^-1, and it is unique.
Answer: A is invertible when another matrix undoes it completely, giving the identity both ways.
Check: Only square matrices can be invertible - the definition needs both products to exist and equal I.
Watch for this: Do not assume every square matrix has an inverse; the next chapter shows exactly which ones do.
Now try it yourself
If A^-1 exists, how many inverses can A have?
Check your answer
Exactly one.
The chapter proves the inverse is unique.
Try a mixed set
Choose a method or explanation before opening each answer.
1. When can A and B be multiplied as AB?
Show answer and reason
When the number of columns of A equals the number of rows of B.
Each entry of AB is a row-column dot product.
2. Is AB always equal to BA?
Show answer and reason
No - matrix multiplication is not commutative in general.
A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]] give AB = [[2, 1], [4, 3]] but BA = [[3, 4], [1, 2]].
3. What is (AB)^T?
Show answer and reason
B^T A^T.
The transpose of a product reverses the order.
4. Diagonal of a skew symmetric matrix?
Show answer and reason
All zeros.
Each diagonal entry would have to equal its own negative.
Try it yourself
Work these on paper before checking any notes. Answers are deliberately not shown here.
- A 3 x 2 matrix multiplies a 2 x 4 matrix. Is the product defined, and what is its order?
- Build a symmetric matrix from M + M^T where M = [[1, 2], [3, 4]].
- Check that [[1, 1], [0, 1]] times [[1, -1], [0, 1]] equals I.
Know the rule but lose track of the steps?
Bring your book edition, your school topic list and a question you attempted. Ask a tutor to explain the choice of method and test it with a different problem. Agree the lesson topics, schedule and fee before starting.
For cost context, see home tuition fees in Delhi NCR. Agree the topics, schedule and fee with your chosen tutor.
Questions about this practice page
- When can two matrices be multiplied?
- Only when the number of columns of the first matrix equals the number of rows of the second. The product then takes the rows of the first and the columns of the second.
- What is the order of a matrix?
- Rows x columns. A matrix with m rows and n columns has order m x n and m times n entries.
- What is the difference between symmetric and skew symmetric matrices?
- A symmetric matrix equals its own transpose; a skew symmetric matrix equals the negative of its transpose and must have a zero diagonal.
- What is an invertible matrix?
- A square matrix A is invertible when some matrix B satisfies AB = BA = I. That B is the unique inverse A^-1.
- Are these the NCERT exercise questions for Matrices (Class 12)?
- No. EZT pages are original practice following the chapter's section order - use them to prepare for the textbook exercises, not as a substitute for your book.