Matrices

Matrices

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

A matrix is a rectangular array of numbers written in m rows and n columns, with order m × n. The entry in row i and column j is denoted a_ij. Two matrices can be added only when their orders match, and multiplication A · B is defined only when the number of columns of A equals the number of rows of B.

For a square matrix A of order n, the inverse is A⁻¹ = (1/det A) · adj A, which exists only when det A ≠ 0. A linear system AX = B is solved by X = A⁻¹B when A is non-singular. For a 2 × 2 matrix A = [[a, b], [c, d]], det A = ad − bc and A⁻¹ = (1/(ad − bc)) [[d, −b], [−c, a]].

High-yield pointers for CUET UG:

• Matrices carry roughly 2–3 questions per attempt and contribute about 3% of the Mathematics paper.
• Watch the property (AB)⁻¹ = B⁻¹A⁻¹ — the order reverses, a classic MCQ trap.
• Cramer’s rule applies only to non-singular systems; do not use it when det A = 0.

---

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Core Definitions and Types

A matrix A = [a_ij] of order m × n has m · n entries. Special types include the square matrix (m = n), diagonal matrix (a_ij = 0 for i ≠ j), identity matrix I_n (diagonal entries 1, all others 0), null matrix (every entry 0), and symmetric / skew-symmetric matrices where Aᵀ = A or Aᵀ = −A respectively.

Operations and Transpose

Addition is entry-wise: (A + B)_ij = a_ij + b_ij, requiring equal orders. Scalar multiplication scales every entry. For product AB, the inner dimensions must match: if A is m × n and B is n × p, the result is m × p with (AB)_ij = Σ a_ik · b_kj. Transposition swaps rows and columns, and obeys (A + B)ᵀ = Aᵀ + Bᵀ along with (AB)ᵀ = BᵀAᵀ — again, the order reverses.

Inverse and Solving AX = B

For a square matrix A, the inverse satisfies AA⁻¹ = A⁻¹A = I and exists only when det A ≠ 0. A non-singular system AX = B has the unique solution X = A⁻¹B, computed efficiently for 2 × 2 by swapping diagonal entries, negating off-diagonals, and dividing by det A.

Cramer’s Rule

For AX = B with det A ≠ 0, each unknown is x_i = det(A_i) / det(A), where A_i is A with its i-th column replaced by B. The rule is fast for n = 2 or 3 but breaks down for singular systems.

Operation	Requirement	Key Property
A + B	Same order m × n	Commutative, associative
cA	Any order	(cA)ᵀ = cAᵀ
AB	Cols of A = Rows of B	(AB)ᵀ = BᵀAᵀ
A⁻¹	Square, det A ≠ 0	(AB)⁻¹ = B⁻¹A⁻¹

Tip: In CUET MCQs, you can often eliminate a wrong option by checking the order of the claimed product or inverse — students lose marks by not verifying dimensions first.

---

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Determinant Expansion and Minors vs Cofactors

For order 3, det A expands along any row or column: det A = a_11(a_22a_33 − a_23a_32) − a_12(a_21a_33 − a_23a_31) + a_13(a_21a_32 − a_22a_31). The minor M_ij is the determinant of the (n − 1) × (n − 1) submatrix obtained by deleting row i and column j, while the cofactor C_ij = (−1)^{i+j} · M_ij carries the sign. Confusing these signs is one of the most frequent calculation errors in CUET numericals.

Rank and Consistency of Linear Systems

The rank of A is the number of non-zero rows in its row-echelon form after elementary operations. For the augmented matrix [A | B]:

• If rank(A) = rank([A | B]) = n, the system has a unique solution.
• If rank(A) = rank([A | B]) < n, the system has infinitely many solutions.
• If rank(A) < rank([A | B]), the system is inconsistent with no solution.

Geometric Applications

The area of a triangle with vertices (x₁, y₁), (x₂, y₂), (x₃, y₃) equals (1/2)|det M| where M has rows (x₁, y₁, 1), (x₂, y₂, 1), (x₃, y₃, 1). Matrices also encode rotations and reflections as linear transformations, useful in coordinate geometry questions.

Special Matrix Classes

An idempotent matrix satisfies A² = A, an involutory matrix satisfies A² = I, and a nilpotent matrix satisfies Aᵏ = 0 for some k. An orthogonal matrix satisfies AᵀA = I, equivalently A⁻¹ = Aᵀ.

Trap alert: Never assume a skew-symmetric matrix has zero determinant — the result holds only for odd orders; for even order, det A can be a perfect square (non-negative). Always verify on the given order.

Practice Prompts

1. If A = [[2, 1], [7, 4]] and B = [[1, 0], [2, 1]], compute AB and verify (AB)⁻¹ = B⁻¹A⁻¹ numerically.
2. Solve the system 2x + 3y = 8, 4x + 5y = 14 using both the inverse method and Cramer’s rule; confirm the result.

---

Continue your study

• View this topic in your CUET UG roadmap — see where “Matrices” fits in your personalised plan
• Build a quick revision plan — 1-day sprint covering highest-weight topics
• CUET UG exam overview — pattern, eligibility, and syllabus
• All Mathematics notes — browse sibling topics in this subject

---

Content adapted based on your selected roadmap duration. Switch tiers using the selector above.