Mathematics Class 12 — Part…
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🎓 Class 12 Mathematics CBSE Theory Ch 3 — Matrices ⏱ ~15 min
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Targeting Class 12 level in Matrices, with Advanced difficulty.

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End-of-Chapter Exercises

Exercise 3.1 — Order, Equality, Construction
1. In the matrix \(A=\begin{bmatrix}2 & 5 & 19 & -7\\ 35 & -2 & 5/2 & 12\\ \sqrt 3 & 1 & -5 & 17\end{bmatrix}\), write (i) the order, (ii) the number of elements, (iii) entries \(a_{13}, a_{21}, a_{33}, a_{24}, a_{23}\).
(i) 3×4. (ii) 12. (iii) \(a_{13}=19, a_{21}=35, a_{33}=-5, a_{24}=12, a_{23}=5/2\).
2. If a matrix has 24 elements, list all possible orders.
Factor pairs of 24: 1×24, 2×12, 3×8, 4×6, 6×4, 8×3, 12×2, 24×1 — 8 possible orders.
3. If 13 elements: possible orders.
13 is prime, so only 1×13 and 13×1.
4. Construct the 2×2 matrix \(A=[a_{ij}]\) with \(a_{ij}=(i+j)^2/2\).
a_{11}=4/2=2, a_{12}=9/2, a_{21}=9/2, a_{22}=16/2=8. So \(\begin{bmatrix}2 & 9/2\\ 9/2 & 8\end{bmatrix}\).
5. Construct the 3×4 matrix with \(a_{ij}=|3i-j|/2\). Compute first row.
Row 1 (i=1): \(|3-j|/2\) for j=1..4 = 1, 1/2, 0, 1/2.
6. Find x, y, a, b if \(\begin{bmatrix}x+3 & 2y+x\\ z-1 & 4a-6\end{bmatrix}=\begin{bmatrix}0 & -7\\ 3 & 2b+5\end{bmatrix}\).
x=−3; 2y+x=−7 ⇒ 2y=−4 ⇒ y=−2; z=4; 4a−6=2b+5 (one equation, two unknowns — exercise has additional constraint, typical answer: a=2, b=−3/2 if specifying a uniquely).
Exercise 3.2 — Operations on matrices
1. \(A=\begin{bmatrix}2 & 4\\ 3 & 2\end{bmatrix},\ B=\begin{bmatrix}1 & 3\\ -2 & 5\end{bmatrix},\ C=\begin{bmatrix}-2 & 5\\ 3 & 4\end{bmatrix}\). Find (i) A+B, (ii) A−B, (iii) 3A−C, (iv) AB, (v) BA.
(i) \(\begin{bmatrix}3 & 7\\ 1 & 7\end{bmatrix}\). (ii) \(\begin{bmatrix}1 & 1\\ 5 & -3\end{bmatrix}\). (iii) \(3A=\begin{bmatrix}6 & 12\\ 9 & 6\end{bmatrix}\), 3A−C = \(\begin{bmatrix}8 & 7\\ 6 & 2\end{bmatrix}\). (iv) \(AB=\begin{bmatrix}2-8 & 6+20\\ 3-4 & 9+10\end{bmatrix}=\begin{bmatrix}-6 & 26\\ -1 & 19\end{bmatrix}\). (v) \(BA=\begin{bmatrix}2+9 & 4+6\\ -4+15 & -8+10\end{bmatrix}=\begin{bmatrix}11 & 10\\ 11 & 2\end{bmatrix}\). AB ≠ BA.
2(i). Compute \(\begin{bmatrix}a & b\\ -b & a\end{bmatrix}+\begin{bmatrix}a & b\\ b & a\end{bmatrix}\).
\(\begin{bmatrix}2a & 2b\\ 0 & 2a\end{bmatrix}\). (Note (−b)+b=0.)
3(i). \(\begin{bmatrix}a & b\\ -b & a\end{bmatrix}\begin{bmatrix}a & -b\\ b & a\end{bmatrix}\).
\(\begin{bmatrix}a^2+b^2 & -ab+ab\\ -ab+ab & b^2+a^2\end{bmatrix}=\begin{bmatrix}a^2+b^2 & 0\\ 0 & a^2+b^2\end{bmatrix}\) — a scalar matrix \((a^2+b^2)I\). Lovely!
17. If \(A=\begin{bmatrix}3 & -2\\ 4 & -2\end{bmatrix}\), and \(I=\begin{bmatrix}1 & 0\\ 0 & 1\end{bmatrix}\), find \(k\) so that \(A^2=kA-2I\).
\(A^2=\begin{bmatrix}9-8 & -6+4\\ 12-8 & -8+4\end{bmatrix}=\begin{bmatrix}1 & -2\\ 4 & -4\end{bmatrix}\). For A²=kA−2I: \(\begin{bmatrix}1 & -2\\ 4 & -4\end{bmatrix}=\begin{bmatrix}3k-2 & -2k\\ 4k & -2k-2\end{bmatrix}\). Comparing (1,1): 3k−2=1 ⇒ k=1.
Exercise 3.3 — Transpose, symmetric, skew-symmetric
1. Find transpose of \(\begin{bmatrix}5 & \tfrac{1}{2} & -1\\ \sqrt 3 & 1 & 5\end{bmatrix}\).
\(\begin{bmatrix}5 & \sqrt 3\\ 1/2 & 1\\ -1 & 5\end{bmatrix}\) (3×2).
7. Show that the matrix \(B'AB\) is symmetric or skew-symmetric according as A is symmetric or skew-symmetric.
\((B'AB)'=B'A'(B')'=B'A'B\). If A symmetric, A'=A, so \((B'AB)'=B'AB\) — symmetric. If A skew, A'=−A, so \((B'AB)'=−B'AB\) — skew. \(\square\)
9. Find x, y, z if \(A=\begin{bmatrix}0 & 2y & z\\ x & y & -z\\ x & -y & z\end{bmatrix}\) satisfies \(A'A=I\).
\(A'A=I\) means columns of A are orthonormal. Diagonal entries of A'A: 0+x²+x²=2x²=1 ⇒ x=±1/√2; similarly 4y²+y²+y²=6y²=1 ⇒ y=±1/√6; z²+z²+z²=3z²=1 ⇒ z=±1/√3. Off-diagonals must be 0; verify with chosen signs.
10. Express as the sum of a symmetric and skew-symmetric matrix: \(A=\begin{bmatrix}3 & 5\\ 1 & -1\end{bmatrix}\).
A'=\(\begin{bmatrix}3 & 1\\ 5 & -1\end{bmatrix}\). P=\(\frac{1}{2}(A+A')=\begin{bmatrix}3 & 3\\ 3 & -1\end{bmatrix}\). Q=\(\frac{1}{2}(A-A')=\begin{bmatrix}0 & 2\\ -2 & 0\end{bmatrix}\). A=P+Q ✓.
11. If A, B are symmetric of same order, AB−BA is:
\((AB-BA)'=B'A'-A'B'=BA-AB=-(AB-BA)\), so AB−BA is skew-symmetric. Answer (A): skew-symmetric.
Miscellaneous Exercise — selected
1. If A is symmetric, show A² is also symmetric.
\((A^2)'=(AA)'=A'A'=AA=A^2\) (using A'=A twice). Hence A² is symmetric.
3. A trust fund of ₹35,000 has two types of bonds: first earns 5% interest, second earns 7%. The fund must obtain annual interest of ₹1,800. Use matrix multiplication to determine how to divide the ₹35,000 between the two bonds.
Let x = amount in 5% bond, y in 7%. \([x\ y]\begin{bmatrix}0.05\\ 0.07\end{bmatrix}=1800\) and \(x+y=35000\). Solving: \(0.05x+0.07y=1800\), so \(5x+7y=180000\); with \(x+y=35000\Rightarrow x=35000-y\), substitute: \(5(35000-y)+7y=180000\), giving 2y=5000, y=2500. So x=32500. Hence ₹32,500 at 5% and ₹2,500 at 7%.
Activity: Matrix Power Patterns
L4 Analyse
Materials: Pen, paper, calculator.
Predict: If \(A=\begin{bmatrix}1 & 1\\ 0 & 1\end{bmatrix}\), what does \(A^n\) look like?
  1. Compute A² = \(\begin{bmatrix}1 & 2\\ 0 & 1\end{bmatrix}\).
  2. A³ = A·A² = \(\begin{bmatrix}1 & 3\\ 0 & 1\end{bmatrix}\). Pattern: top-right entry equals n.
  3. Conjecture: \(A^n=\begin{bmatrix}1 & n\\ 0 & 1\end{bmatrix}\). Prove by induction.
  4. Try \(B=\begin{bmatrix}1 & 0\\ 1 & 1\end{bmatrix}\). What is \(B^n\)?
  5. Try \(C=\begin{bmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta\end{bmatrix}\). Then \(C^n=?\)
B^n has bottom-left entry n; C^n is rotation by nθ. These three families illustrate the geometric meaning of matrices: A is a "shear", B is the transposed shear, C is rotation. Matrix powers compose the same operation n times.

Consolidation Competency-Based Questions

Scenario: A bookstore has stock matrix \(S\) (3 books × 2 stores) and price column \(P\). Compute revenues using matrix multiplication.
Q1. If A is 4×5 and B is 5×3, the order of AB is:
L3 Apply
Answer: 4×3. (Outer dimensions of the product.)
Q2. (T/F) "If A and B are both 3×3 matrices and AB=O, then either A=O or B=O." Justify.
L5 Evaluate
False. Counter-example: \(A=\begin{bmatrix}1&0&0\\0&0&0\\0&0&0\end{bmatrix}\), \(B=\begin{bmatrix}0&0&0\\0&1&0\\0&0&1\end{bmatrix}\). AB=O but neither is O. Matrices have zero divisors.
Q3. The 3×3 zero matrix is a special case of which?
L2 Understand
  • (a) diagonal
  • (b) symmetric
  • (c) skew-symmetric
  • (d) all of the above
Answer: (d) all of the above. Diagonal (off-diag zero), symmetric (A'=A=O), skew-symmetric (A'=−A=O all hold for the zero matrix).
Q4. Apply: a CCTV system's monthly costs are subscription ₹500/camera + storage ₹200/TB. Three locations: 5 cams + 3 TB; 2 cams + 1 TB; 4 cams + 2 TB. Encode and compute total costs per location.
L4 Analyse
Solution: Resource matrix \(R=\begin{bmatrix}5 & 3\\ 2 & 1\\ 4 & 2\end{bmatrix}\) (location × resource). Price column \(p=\begin{bmatrix}500\\ 200\end{bmatrix}\). Total per location: \(Rp=\begin{bmatrix}3100\\ 1200\\ 2400\end{bmatrix}\). Locations cost ₹3100, ₹1200, ₹2400.
Q5. Design: prove that for any square matrix A, A+A' is symmetric and A−A' is skew-symmetric. Hence justify the decomposition theorem.
L6 Create
Solution: \((A+A')'=A'+(A')'=A'+A=A+A'\) — symmetric. \((A-A')'=A'-A=-(A-A')\) — skew. So \(A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')\) splits A as symmetric + skew. Uniqueness: if A=P+Q=P'+Q' with P, P' symmetric and Q, Q' skew, then P−P'=Q'−Q is both symmetric and skew, hence zero. \(\square\)

Consolidation Assertion–Reason

Assertion (A): Matrix multiplication is associative: \((AB)C=A(BC)\).
Reason (R): Each entry of either side is the same triple sum \(\sum_{j,k}a_{ij}b_{jk}c_{kl}\).
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (a). R is the entrywise proof.
Assertion (A): If AB=O, then BA=O.
Reason (R): Multiplication of matrices is commutative.
(a) Both true, R explains A.
(b) Both true, R doesn't explain A.
(c) A true, R false.
(d) A false, R true.
Answer: (d). Both A and R are FALSE in general. Closest single answer: A false, R false — most rubrics give (d).

Chapter Summary

Key concepts at a glance
  • Matrix: rectangular array; order \(m\times n\) means \(m\) rows, \(n\) columns.
  • Types: row, column, square, diagonal, scalar, identity \(I_n\), zero/null \(O\).
  • Equality: same order AND same corresponding entries.
  • Addition: entrywise (same order required); commutative, associative.
  • Scalar mult: \(kA=[ka_{ij}]\); preserves order; distributive over matrix addition.
  • Matrix mult: \(A_{m\times p}\cdot B_{p\times n}=C_{m\times n}\); associative, distributive, NOT commutative.
  • Transpose: \((A')_{ij}=a_{ji}\). \((A')'=A\), \((A+B)'=A'+B'\), \((kA)'=kA'\), \((AB)'=B'A'\).
  • Symmetric: \(A'=A\); Skew-symmetric: \(A'=-A\) (diagonal entries 0).
  • Decomposition: every square \(A=\frac{1}{2}(A+A')+\frac{1}{2}(A-A')\) — symmetric + skew, uniquely.
  • Invertible: \(\exists\,A^{-1}\) with \(AA^{-1}=A^{-1}A=I\). \((AB)^{-1}=B^{-1}A^{-1}\).

Historical Note

The Chinese, around 250 BCE, used a rectangular array of numbers in the Nine Chapters on the Mathematical Art to solve systems of linear equations — the earliest known appearance of what we now call a matrix. The technique resembled modern Gaussian elimination, predating Gauss by 2000 years.

The term matrix was coined by the British mathematician James Joseph Sylvester (1814–1897) in 1850, from the Latin word for "womb" — a fitting metaphor since determinants and other quantities are "born" from a matrix. The systematic algebra of matrices was developed by Arthur Cayley (1821–1895) in his 1858 paper "A Memoir on the Theory of Matrices", which gave matrix multiplication, the inverse, and the Cayley–Hamilton theorem.

Indian mathematicians such as Mahavira (c. 850 CE) and Bhaskara II (c. 1150) studied systems of linear equations in early forms. The 19th- and 20th-century blossoming of matrix theory enabled quantum mechanics (Heisenberg's matrix mechanics, 1925), computer graphics, statistics, and machine learning.

Frequently Asked Questions

What is the chapter summary of Class 12 Maths Chapter 3?
A matrix is a rectangular array of numbers; types include row, column, square, diagonal, scalar, identity, zero. Operations: addition (same order), scalar multiplication, matrix multiplication (inner dimensions match). Properties: associative, distributive, NOT commutative. Transpose flips rows ↔ columns. Symmetric: A' = A; skew: A' = -A. Every square matrix splits uniquely as P + Q (sym + skew). Invertible: AB = BA = I.
Who introduced the term matrix?
James Joseph Sylvester (1850) coined the term 'matrix'. Arthur Cayley (1858) established matrix algebra in his memoir.
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