Why this chapter is mostly about what fails
Students arrive at matrices with a decade of ordinary algebra behind them, and most of it transfers. Addition is commutative and associative, scalar multiplication distributes, and the identity behaves as expected. The exam concentrates on the handful of places where the transfer breaks.
There are essentially four: multiplication does not commute, products can vanish without either factor vanishing, expansions like (A + B)² require AB = BA, and transposing or inverting a product reverses the order. A student who has internalised those four handles the conceptual questions; one who has not will get them wrong confidently.
How JEE actually asks Matrices
NTA publishes no chapter-wise weightage, so figures online are coaching estimates from past papers; check the official NTA information bulletin for your session's syllabus and pattern.
Matrices is usually asked together with Determinants, and the pair forms a compact, predictable block. Questions divide between computation - multiply these, find this inverse - and property checks, which are the conceptual failures listed above. Systems of linear equations bridge the two chapters and appear regularly.
Key concepts, compressed
- ✓A matrix of order m × n has m rows and n columns. Addition requires identical orders.
- ✓AB is defined only when the number of columns of A equals the number of rows of B, and the result is rows(A) × cols(B).
- ✓Matrix multiplication is associative and distributive but not commutative.
- ✓The transpose reflects a matrix across its main diagonal; symmetric means Aᵀ = A and skew-symmetric means Aᵀ = −A.
- ✓The inverse satisfies AA⁻¹ = I and exists only when |A| ≠ 0, in which case A⁻¹ = adj(A)/|A|.
- ✓A system AX = B has a unique solution when |A| ≠ 0, given by X = A⁻¹B.
Results you need before attempting the questions
| Property | Result | Note |
|---|---|---|
| Product defined | (m × n)(n × p) = m × p | inner dimensions must match |
| Commutativity | AB ≠ BA in general | |
| Transpose of a product | (AB)ᵀ = BᵀAᵀ | order reverses |
| Inverse of a product | (AB)⁻¹ = B⁻¹A⁻¹ | order reverses |
| Inverse | A⁻¹ = adj(A)/|A| | needs |A| ≠ 0 |
| Adjoint property | A · adj(A) = |A| I | |
| Determinant of adjoint | |adj A| = |A|^(n−1) | n is the order |
| Symmetric | Aᵀ = A | |
| Skew-symmetric | Aᵀ = −A | diagonal entries all zero |
| Decomposition | A = ½(A + Aᵀ) + ½(A − Aᵀ) | unique |
| System of equations | AX = B ⟹ X = A⁻¹B | if |A| ≠ 0 |
| Homogeneous system | non-trivial solution iff |A| = 0 |
The five mistakes that cost the most marks
- ✓Assuming AB = BA. They are usually different matrices, and swapping them mid-solution invalidates everything after.
- ✓Concluding from AB = 0 that A or B must be zero. Non-zero matrices can multiply to the zero matrix.
- ✓Writing (AB)ᵀ = AᵀBᵀ or (AB)⁻¹ = A⁻¹B⁻¹. Both operations reverse the order of the factors.
- ✓Inverting without checking the determinant. A singular matrix has no inverse, and parameter questions usually turn on the value that makes |A| = 0.
- ✓Expanding (A + B)² as A² + 2AB + B². That requires AB = BA; in general it is A² + AB + BA + B².
Practice set 1: orders and multiplication
1. What is the order of a matrix with 3 rows and 4 columns, and how many elements does it have?
Order 3 × 4, with 12 elements. The convention is always rows first, then columns - reversing it is a small error that propagates into every subsequent dimension check. The number of elements is simply the product of the two.
2. Under what condition can two matrices be added?
They must have exactly the same order. Addition is elementwise, so there must be a corresponding element for every position. This is a stricter condition than for multiplication, which only requires the inner dimensions to agree - a distinction questions test directly.
3. If A is 2 × 3 and B is 3 × 4, is AB defined? What is its order?
Yes, and AB is 2 × 4. The product is defined when the number of columns of A equals the number of rows of B - here both are 3 - and the result takes the outer dimensions. Note that BA is not defined at all, since B has 4 columns and A has 2 rows.
4. Is matrix multiplication commutative?
No. In general AB ≠ BA, and as question 3 shows, one product can be defined while the other is not. Even for square matrices of the same order, the two products are usually different matrices. This single fact underlies most of the chapter's conceptual questions.
5. For A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], compute AB and BA.
AB = [[2, 1], [4, 3]] and BA = [[3, 4], [1, 2]]. For AB, the first row is [1(0) + 2(1), 1(1) + 2(0)] = [2, 1] and the second is [3(0) + 4(1), 3(1) + 4(0)] = [4, 3]. For BA, the first row is [0(1) + 1(3), 0(2) + 1(4)] = [3, 4] and the second is [1, 2]. The two results are clearly different, which is the point of the example - B here swaps columns on the right and rows on the left.
6. Simplify (AB)ᵀ.
BᵀAᵀ - the order reverses. Writing AᵀBᵀ is the standard error, and it usually is not even a defined product when the matrices are not square. The same reversal applies to inverses: (AB)⁻¹ = B⁻¹A⁻¹.
7. What is the identity matrix, and what does it do?
A square matrix with ones on the main diagonal and zeros elsewhere, satisfying AI = IA = A for any conformable A. It is the multiplicative identity, and it is one of the few matrices that commutes with everything - which is exactly why the inverse is defined by AA⁻¹ = I.
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Practice set 2: special matrices
8. What is a symmetric matrix?
One satisfying Aᵀ = A, so aᵢⱼ = aⱼᵢ for all entries - the matrix is unchanged by reflection across its main diagonal. Symmetric matrices must be square, and the diagonal entries are unconstrained. Both A + Aᵀ and AAᵀ are symmetric for any A, which is a result worth knowing.
9. What is special about the diagonal entries of a skew-symmetric matrix?
They are all zero. Skew-symmetry means Aᵀ = −A, so aᵢⱼ = −aⱼᵢ. Setting i = j gives aᵢᵢ = −aᵢᵢ, which forces aᵢᵢ = 0. This is asked directly, and it also explains why a skew-symmetric matrix of odd order has determinant zero.
10. Can every square matrix be written as the sum of a symmetric and a skew-symmetric matrix?
Yes, and uniquely: A = ½(A + Aᵀ) + ½(A − Aᵀ). The first bracket is symmetric because transposing it leaves it unchanged; the second is skew-symmetric because transposing it negates it. Their sum is clearly A. This decomposition is asked often enough to be worth memorising in that exact form.
11. What distinguishes a diagonal matrix from a scalar matrix?
A diagonal matrix has zeros everywhere off the main diagonal, with the diagonal entries arbitrary. A scalar matrix is a diagonal matrix whose diagonal entries are all equal - so it is a scalar multiple of the identity. Every scalar matrix is diagonal, but not conversely, and scalar matrices commute with everything.
12. What is an orthogonal matrix?
A square matrix satisfying AAᵀ = I, which means A⁻¹ = Aᵀ. Its rows form a set of mutually perpendicular unit vectors, and so do its columns. Taking determinants of AAᵀ = I gives |A|² = 1, so an orthogonal matrix has determinant ±1 - a result questions sometimes ask for directly.
13. What is the trace of a matrix, and what property does it have?
The sum of the entries on the main diagonal, defined only for square matrices. Its useful property is that trace(AB) = trace(BA), even though AB and BA are generally different matrices. It is also additive: trace(A + B) = trace(A) + trace(B).
Practice set 3: determinant and inverse
14. Find the determinant of A = [[1, 2], [3, 4]].
−2. For a 2 × 2 matrix the determinant is ad − bc = (1)(4) − (2)(3) = 4 − 6 = −2. Because it is non-zero, A is non-singular and therefore invertible - which is the check that must precede the next question.
15. Find A⁻¹ for that matrix.
A⁻¹ = [[−2, 1], [1.5, −0.5]]. For a 2 × 2 matrix, adj(A) swaps the diagonal entries and negates the off-diagonal ones, giving [[4, −2], [−3, 1]]. Dividing by |A| = −2 gives [[−2, 1], [1.5, −0.5]]. Verify by multiplying: A A⁻¹ = [[−2 + 3, 1 − 1], [−6 + 6, 3 − 2]] = [[1, 0], [0, 1]], the identity.
16. How should you check an inverse you have computed?
Multiply it by the original matrix and confirm you get the identity. This takes a few seconds for a 2 × 2 and catches every sign and division error. It is the matrix equivalent of differentiating an integral to check it, and it is worth doing every time.
17. When does a square matrix fail to have an inverse?
When its determinant is zero - such a matrix is called singular. Since A⁻¹ = adj(A)/|A|, a zero determinant makes the expression undefined. Parameter questions almost always turn on finding the value that makes |A| = 0, so computing the determinant first is the right opening move.
18. Simplify (AB)⁻¹.
B⁻¹A⁻¹, with the order reversed. Check it: (AB)(B⁻¹A⁻¹) = A(BB⁻¹)A⁻¹ = AIA⁻¹ = AA⁻¹ = I. Writing A⁻¹B⁻¹ instead is the standard error, and the check above shows immediately why it fails - the inner factors would not cancel.
19. For a 3 × 3 matrix A, what is |adj A| in terms of |A|?
|A|², since |adj A| = |A|^(n−1) with n = 3. The general result follows from taking determinants of A · adj(A) = |A| I: the left side gives |A||adj A| and the right gives |A|ⁿ. For a 2 × 2 matrix the exponent is 1, so |adj A| = |A|.
20. What is A · adj(A) equal to?
|A| I - the determinant times the identity matrix. This single relation is where the inverse formula comes from: dividing both sides by |A| gives A(adj A/|A|) = I, identifying adj(A)/|A| as the inverse. It also shows why a zero determinant breaks the construction.
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Practice set 4: systems of linear equations
21. How do you write a system of linear equations in matrix form and solve it?
As AX = B, where A holds the coefficients, X the variables and B the constants. If A is invertible, multiply on the left by A⁻¹ to get X = A⁻¹B. Note that it must be on the left - multiplying on the right would give XA⁻¹, which is a different and generally undefined product.
22. When does a system of linear equations have a unique solution?
When the coefficient matrix is non-singular, that is |A| ≠ 0. Then A⁻¹ exists and X = A⁻¹B gives exactly one answer. Geometrically, for three equations in three unknowns, it means the three planes meet at a single point.
23. What happens when |A| = 0?
The system has either no solution or infinitely many - never exactly one. Which case applies depends on whether the constants are consistent with the dependency among the equations, and it is distinguished using the determinants from Cramer's rule or by row reduction. Geometrically the planes are either parallel or intersect in a line or plane.
24. When does a homogeneous system AX = 0 have a non-trivial solution?
Exactly when |A| = 0. A homogeneous system always has the trivial solution X = 0, so the question is only ever whether others exist. If A is invertible then X = A⁻¹0 = 0 is the only solution, so non-trivial solutions require A to be singular.
25. What does it mean for a system to be consistent?
That at least one solution exists. A consistent system may have one solution or infinitely many; an inconsistent system has none. Note that a homogeneous system is always consistent, since X = 0 always works - so questions about inconsistency only arise for non-homogeneous systems.
Practice set 5: properties that fail
26. If AB = 0, does it follow that A = 0 or B = 0?
No. Take A = [[1, 0], [0, 0]] and B = [[0, 0], [0, 1]]: neither is the zero matrix, yet AB = 0. Matrices have zero divisors, unlike real numbers. A consequence is that you cannot cancel matrices in an equation - from AB = AC you cannot conclude B = C unless A is invertible.
27. Expand (A + B)².
A² + AB + BA + B², which equals A² + 2AB + B² only when AB = BA. Since matrix multiplication does not commute in general, the familiar binomial expansion is invalid. This is one of the most frequently tested traps in the chapter, and the correct four-term form is what to write.
28. Simplify (Aᵀ)ᵀ and (A + B)ᵀ.
(Aᵀ)ᵀ = A and (A + B)ᵀ = Aᵀ + Bᵀ. Transposition is its own inverse, and it distributes over addition without any reversal. The reversal only appears for products - which is exactly why questions pair these with (AB)ᵀ to see whether you apply the rule indiscriminately.
29. What is the rank of a matrix?
The order of the largest non-zero minor, equivalently the number of linearly independent rows or columns. A square matrix of order n is invertible precisely when its rank is n. Rank is the tool for deciding consistency when the determinant is zero and Cramer's rule cannot distinguish the cases.
30. What are idempotent and nilpotent matrices?
An idempotent matrix satisfies A² = A, and a nilpotent matrix satisfies Aᵏ = 0 for some positive integer k. Both are impossible for non-zero real numbers other than 0 and 1, which is why they appear only in matrix algebra. A nilpotent matrix necessarily has determinant zero, since |A|ᵏ = 0 forces |A| = 0.
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How to study this chapter efficiently
- ✓Write out the dimensions before multiplying anything. Half the errors in this chapter are products that were never defined.
- ✓Learn the four failures as a block: no commutativity, zero divisors exist, binomial expansion needs commutativity, and transpose and inverse reverse order.
- ✓Compute the determinant before attempting any inverse. If it is zero, the question is about singularity, not inversion.
- ✓Check every inverse by multiplying back to the identity. It takes seconds and catches all sign errors.
- ✓Memorise the symmetric and skew-symmetric decomposition in the exact form ½(A + Aᵀ) + ½(A − Aᵀ).
- ✓Study Matrices and Determinants together. They share the inverse, the adjoint and the linear-systems material, so preparing them separately duplicates work.
Turn this into active practice
The property questions in this chapter are where marks are actually decided, and they are exactly the ones that feel obvious when you read them and catch you when you answer them under time pressure. Knowing that AB ≠ BA is not the same as remembering it while simplifying a four-factor expression.
The JEE Matrices quiz on QUFF generates fresh questions across multiplication, special matrices, inverses, linear systems and the property traps, marks them instantly and explains each answer. Do a mixed set and check whether your errors were arithmetic or property-based - the second kind repeats until you name it.
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