Mathematics Class 12 — Part…
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3.1 Introduction

🎓 Class 12 Mathematics CBSE Theory Ch 3 — Matrices ⏱ ~15 min
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This MCQ module is based on: 3.1 Introduction

This mathematics assessment will be based on: 3.1 Introduction
Targeting Class 12 level in Matrices, with Advanced difficulty.

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3.1 Introduction

The knowledge of matrices? is necessary in various branches of mathematics. They give a compact way to write a system of linear equations, store data tables (sales, populations, image pixels), describe rotations in computer graphics, and encode the link structure of the web (PageRank). The same algebra repeats from physics to economics: a matrix is the natural object whenever you need a rectangular block of numbers and a way to combine them.

Imagine Radha's notebook listing 15 notebooks at ₹50 and 6 pens at ₹15. As a matrix: \[\begin{bmatrix}15 & 6\end{bmatrix}\quad\text{or vertically}\quad\begin{bmatrix}15\\ 6\end{bmatrix}.\] Combining her record with Fauzia's gives a 2×2 matrix. The chapter develops a general algebra of these objects.

3.2 Matrix

Definition: Matrix
A matrix is a rectangular array of numbers (or functions, or any expressions) arranged in rows (horizontal) and columns (vertical). An \(m\times n\) matrix has \(m\) rows and \(n\) columns, total \(mn\) entries: \[A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{m1}&a_{m2}&\cdots&a_{mn}\end{bmatrix}=[a_{ij}]_{m\times n}.\] Here \(a_{ij}\) is the entry in row \(i\), column \(j\). The pair \((m,n)\) is the order of the matrix.

3.2.1 Order of a matrix

An \(m\times n\) matrix has \(mn\) entries. Hence to construct a matrix you choose a pair \((m,n)\) with \(mn=N\) for some required total \(N\). For example, a matrix with 12 entries can have orders \(1\times 12,\ 2\times 6,\ 3\times 4,\ 4\times 3,\ 6\times 2,\ 12\times 1\) — six choices.

3.3 Types of Matrices

Common matrix types
  1. Column matrix: single column, order \(m\times 1\).
  2. Row matrix: single row, order \(1\times n\).
  3. Square matrix: equal rows and columns, order \(n\times n\).
  4. Diagonal matrix: square with \(a_{ij}=0\) for \(i\ne j\) (only diagonal entries non-zero).
  5. Scalar matrix: diagonal matrix with all diagonal entries equal: \(a_{ii}=k\) for some constant \(k\).
  6. Identity matrix \(I_n\): scalar matrix with \(k=1\). \(I_n=[\delta_{ij}]\) where \(\delta_{ii}=1,\ \delta_{ij}=0\) (\(i\ne j\)).
  7. Zero matrix \(O\): every entry is 0.
Diagonal: [ 200 050 007 ] Scalar: [ 300 030 003 ] Identity I₃: [ 100 010 001 ]

3.4 Equality of Matrices

Equality
Two matrices \(A=[a_{ij}]\) and \(B=[b_{ij}]\) are equal iff
  1. they have the same order, AND
  2. \(a_{ij}=b_{ij}\) for every pair \((i,j)\).

3.5 Operations on Matrices

3.5.1 Addition of matrices

Addition
For matrices \(A,B\) of the same order \(m\times n\), define \[(A+B)_{ij}=a_{ij}+b_{ij}.\] The sum has the same order as the addends. Addition is undefined for matrices of different orders.

Properties (proved entry-wise from real-number axioms):

  • Commutative: \(A+B=B+A\).
  • Associative: \((A+B)+C=A+(B+C)\).
  • Additive identity: \(A+O=A\) where \(O\) is the zero matrix of the same order.
  • Additive inverse: \(A+(-A)=O\) where \((-A)_{ij}=-a_{ij}\).

3.5.2 Multiplication by a scalar

Scalar multiplication
For a scalar \(k\in\mathbb R\) (or \(\mathbb C\)) and a matrix \(A=[a_{ij}]\), \[(kA)_{ij}=k\cdot a_{ij}.\] Multiply every entry by \(k\). The order is preserved.

Properties:

  • \(k(A+B)=kA+kB\) (scalar distributes over addition).
  • \((k+l)A=kA+lA\).
  • \(k(lA)=(kl)A\).
  • \(1\cdot A=A,\quad 0\cdot A=O\).

Worked Examples

Example 1. Consider \(A=\begin{bmatrix}1&2&3\\ 4&5&6\end{bmatrix}\). State its order, and identify \(a_{12},\ a_{23},\ a_{21}\).
Order \(2\times 3\). \(a_{12}=2\), \(a_{23}=6\), \(a_{21}=4\).
Example 2. If \(\begin{bmatrix}x+3 & z+4 & 2y-7\\ -6 & a-1 & 0\\ b-3 & -21 & 0\end{bmatrix}=\begin{bmatrix}0 & 6 & 3y-2\\ -6 & -3 & 2c+2\\ 2b+4 & -21 & 0\end{bmatrix}\), find \(a, b, c, x, y, z\).
Equate corresponding entries:
\(x+3=0\Rightarrow x=-3\); \(z+4=6\Rightarrow z=2\); \(2y-7=3y-2\Rightarrow y=-5\).
\(a-1=-3\Rightarrow a=-2\); \(0=2c+2\Rightarrow c=-1\); \(b-3=2b+4\Rightarrow b=-7\).
Example 3. Compute \(A+B\) where \(A=\begin{bmatrix}\sqrt 3 & 1 & -1\\ 2 & 3 & 0\end{bmatrix}\), \(B=\begin{bmatrix}2\sqrt 3 & 5 & 3\\ -1 & 0 & 4\end{bmatrix}\).
\(A+B=\begin{bmatrix}3\sqrt 3 & 6 & 2\\ 1 & 3 & 4\end{bmatrix}\). (Add entrywise.)
Example 4. Find \(3A-B\) where \(A=\begin{bmatrix}1 & 2 & -3\\ 5 & 0 & 2\\ 1 & -1 & 1\end{bmatrix}\), \(B=\begin{bmatrix}3 & -1 & 2\\ 4 & 2 & 5\\ 2 & 0 & 3\end{bmatrix}\).
\(3A=\begin{bmatrix}3 & 6 & -9\\ 15 & 0 & 6\\ 3 & -3 & 3\end{bmatrix}\); \(3A-B=\begin{bmatrix}0 & 7 & -11\\ 11 & -2 & 1\\ 1 & -3 & 0\end{bmatrix}\).
Example 5. Find X and Y if \(X+Y=\begin{bmatrix}5 & 2\\ 0 & 9\end{bmatrix}\) and \(X-Y=\begin{bmatrix}3 & 6\\ 0 & -1\end{bmatrix}\).
Add: \(2X=\begin{bmatrix}8 & 8\\ 0 & 8\end{bmatrix}\Rightarrow X=\begin{bmatrix}4 & 4\\ 0 & 4\end{bmatrix}\). Subtract: \(2Y=\begin{bmatrix}2 & -4\\ 0 & 10\end{bmatrix}\Rightarrow Y=\begin{bmatrix}1 & -2\\ 0 & 5\end{bmatrix}\).
Activity: Match the Matrix Type
L3 Apply
Materials: Pen, paper.
Predict: Is \(\begin{bmatrix}2 & 0\\ 0 & 2\end{bmatrix}\) a diagonal matrix? An identity? A scalar?
  1. Classify the following matrices: \(\begin{bmatrix}1\\3\\5\end{bmatrix}\) (column), \(\begin{bmatrix}2&3&5\end{bmatrix}\) (row), \(\begin{bmatrix}1&0\\0&1\end{bmatrix}\) (identity), \(\begin{bmatrix}5&0\\0&5\end{bmatrix}\) (scalar), \(\begin{bmatrix}3&0\\0&7\end{bmatrix}\) (diagonal but not scalar).
  2. Note: every identity matrix is a scalar matrix (with k=1); every scalar matrix is a diagonal matrix; not every diagonal matrix is scalar.
  3. Construct: a 2×2 matrix that is diagonal but not scalar, and a 3×3 scalar matrix that is not the identity.
The hierarchy: zero ⊂ scalar (k=0); identity ⊂ scalar (k=1); scalar ⊂ diagonal ⊂ square. Each level adds restrictions, narrowing the family. Recognising the type quickly speeds up future operations like inverse and determinant.

Competency-Based Questions

Scenario: A textile shop's monthly sales are tracked in a 3×4 matrix where rows = product (shirt, trouser, jacket) and columns = month (Jan, Feb, Mar, Apr).
Q1. Total entries in this matrix:
L1 Remember
Answer: \(3\times 4=12\) entries.
Q2. (Fill in) The entry \(a_{23}\) represents the sales of ____ in ____ .
L2 Understand
Answer: Row 2 = Trouser, Column 3 = Mar. So \(a_{23}\) is March trouser sales.
Q3. (T/F) "If A and B both have order 3×4, A+B is defined and has order 3×4." Justify.
L5 Evaluate
True. Addition requires identical orders; the result has the same order.
Q4. The identity matrix \(I_3\) is a special case of which broader category? Choose ALL that apply.
L3 Apply
  • (a) diagonal
  • (b) scalar
  • (c) zero
  • (d) symmetric
Answer: (a), (b), (d). \(I_3\) is diagonal (off-diagonals zero), scalar (all diagonals equal 1), and symmetric (we'll meet in Part 2). It is NOT zero.
Q5. Design: write the 3×4 sales matrix knowing Jan sales (10, 8, 5), Feb (12, 9, 4), Mar (15, 11, 7), Apr (20, 14, 10). Then compute the average monthly sales matrix.
L6 Create
Solution: \(S=\begin{bmatrix}10 & 12 & 15 & 20\\ 8 & 9 & 11 & 14\\ 5 & 4 & 7 & 10\end{bmatrix}\). Average = (1/4)·sum of columns = column matrix \(\begin{bmatrix}14.25\\ 10.5\\ 6.5\end{bmatrix}\). Total entries treated as scalar multiplication of average vector.

Assertion–Reason Questions

Assertion (A): A 2×3 matrix has 6 entries.
Reason (R): The total number of entries in an m×n matrix is m·n.
(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 rule that gives A.
Assertion (A): Two matrices of different orders cannot be equal.
Reason (R): Equality requires same order AND same corresponding entries.
(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 subsumes A: failing the order condition immediately invalidates equality.
Assertion (A): Matrix addition is commutative.
Reason (R): Each entry of A+B is a sum of real numbers, which 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: (a). Entrywise commutativity lifts to the whole matrix.

Frequently Asked Questions

What is a matrix?
A matrix is a rectangular arrangement of numbers (or expressions) in rows and columns. A matrix with m rows and n columns is said to be of order m × n.
What are the types of matrices?
Common types: row (1×n), column (m×1), square (n×n), diagonal, scalar, identity, zero/null.
When are two matrices equal?
Two matrices are equal iff they have the same order AND every corresponding entry is equal.
How do you add two matrices?
Addition is defined only when both have the same order. Add corresponding entries.
What is scalar multiplication of a matrix?
Multiply every entry by the scalar k. Order is preserved.
What is the order of a matrix?
If a matrix has m rows and n columns, its order is m × n. Total entries = m·n.
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