Matrices Operations Class 12 Matrices Tutorial

Matrices Operations | Class 12 Matrices Tutorial

Mathematics Study Notes: Matrices | Class XII

📐 Mathematics Study Notes

Class XII | Chapter: Matrices
✨ Topic: Algebraic Operations and Multiplication Properties ✨
🔷 Concept 1: Basic Operations (Addition and Scalar Multiplication)

Matrix addition and scalar multiplication are the fundamental building blocks of matrix algebra. These operations follow specific conformability rules.

➕ 1. Addition of Matrices

Two matrices \(A\) and \(B\) can be added if and only if they are of the same order. If \(A = [a_{ij}]_{m \times n}\) and \(B = [b_{ij}]_{m \times n}\), then \(A + B = [a_{ij} + b_{ij}]_{m \times n}\).

  • Commutative Law: \(A + B = B + A\)
  • Associative Law: \((A + B) + C = A + (B + C)\)
  • Additive Identity: \(A + O = A\), where \(O\) is the zero matrix.
✖️ 2. Scalar Multiplication

If \(A = [a_{ij}]\) is a matrix and \(k\) is a scalar, then \(kA\) is obtained by multiplying each element by \(k\): \(kA = [k \cdot a_{ij}]\).

  • Distributive Law: \(k(A + B) = kA + kB\) and \((k + l)A = kA + lA\).
📘 Solved Examples (Concept 1)
Example 1:

If \(A = \begin{bmatrix} 2 & 4 \\ 3 & 2 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 3 \\ -2 & 5 \end{bmatrix}\), find \(3A – B\).

Solution: \(3A = \begin{bmatrix} 6 & 12 \\ 9 & 6 \end{bmatrix}\). Then \(3A – B = \begin{bmatrix} 5 & 9 \\ 11 & 1 \end{bmatrix}\).
Example 2:

Find \(X\) and \(Y\) if \(X + Y = \begin{bmatrix} 7 & 0 \\ 2 & 5 \end{bmatrix}\) and \(X – Y = \begin{bmatrix} 3 & 0 \\ 0 & 3 \end{bmatrix}\).

Solution: Adding: \(2X = \begin{bmatrix} 10 & 0 \\ 2 & 8 \end{bmatrix} \Rightarrow X = \begin{bmatrix} 5 & 0 \\ 1 & 4 \end{bmatrix}\), \(2Y = \begin{bmatrix} 4 & 0 \\ 2 & 2 \end{bmatrix} \Rightarrow Y = \begin{bmatrix} 2 & 0 \\ 1 & 1 \end{bmatrix}\).
Example 3:

Find \(x\) and \(y\) from: \(2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} + \begin{bmatrix} 3 & -4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}\).

Solution: \(\begin{bmatrix} 2x+3 & 6 \\ 15 & 2y-4 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix} \Rightarrow x=2,\; y=9\).
Example 4:

If \(A = \operatorname{diag}(1, -1, 2)\) and \(B = \operatorname{diag}(2, 3, -1)\), find \(A+B\) and \(3A\).

Solution: \(A+B = \operatorname{diag}(3,2,1)\), \(3A = \operatorname{diag}(3,-3,6)\).
Example 5:

Simplify: \(\cos \theta \begin{bmatrix} \cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{bmatrix} + \sin \theta \begin{bmatrix} \sin \theta & -\cos \theta \\ \cos \theta & \sin \theta \end{bmatrix}\).

Solution: Result = \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I\).
📌 Board Exam Style (3×3 Matrices)
Example 6:

If \(A = \begin{bmatrix} 1 & 2 & 3 \\ 0 & 5 & 7 \\ 6 & 8 & 9 \end{bmatrix}\) and \(B = \begin{bmatrix} 2 & 0 & 3 \\ 3 & 0 & 5 \\ 5 & 7 & 0 \end{bmatrix}\), find matrix \(C\) such that \(2A + 3B – C = I\).

Solution: \(C = 2A+3B – I\). \(2A = \begin{bmatrix} 2 & 4 & 6 \\ 0 & 10 & 14 \\ 12 & 16 & 18 \end{bmatrix}\), \(3B = \begin{bmatrix} 6 & 0 & 9 \\ 9 & 0 & 15 \\ 15 & 21 & 0 \end{bmatrix}\). Sum = \(\begin{bmatrix} 8 & 4 & 15 \\ 9 & 10 & 29 \\ 27 & 37 & 18 \end{bmatrix}\). Subtract \(I\): \(C = \begin{bmatrix} 7 & 4 & 15 \\ 9 & 9 & 29 \\ 27 & 37 & 17 \end{bmatrix}\).
Example 7:

If \(A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & 1 \\ 2 & 0 & 3 \end{bmatrix}\), prove \(A^3 – 6A^2 + 7A + 2I = O\).

Solution: \(A^2 = \begin{bmatrix} 5 & 0 & 8 \\ 2 & 4 & 5 \\ 8 & 0 & 13 \end{bmatrix}\), \(A^3 = A^2\cdot A = \begin{bmatrix} 21 & 0 & 34 \\ 12 & 8 & 23 \\ 34 & 0 & 55 \end{bmatrix}\). Substituting yields zero matrix.
Example 8:

Solve \(3\begin{bmatrix} x & y \\ z & w \end{bmatrix} = \begin{bmatrix} x & 6 \\ -1 & 2w \end{bmatrix} + \begin{bmatrix} 4 & x+y \\ z+w & 3 \end{bmatrix}\).

Solution: Comparing entries: \(x=2,\; y=4,\; w=3,\; z=1\).
Example 9:

Let \(A = \begin{bmatrix} 0 & c & -b \\ -c & 0 & a \\ b & -a & 0 \end{bmatrix}\), \(B = \begin{bmatrix} a^2 & ab & ac \\ ab & b^2 & bc \\ ac & bc & c^2 \end{bmatrix}\). Find \(AB\).

Solution: Each entry cancels out: \(AB = O\). Non-zero matrices can yield zero product.
Example 10:

If \(A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{bmatrix}\), prove \(A^n = 3^{n-1}A\) for \(n \in \mathbb{N}\).

Solution: By induction: \(A^2 = 3A\). Assume \(A^k = 3^{k-1}A\), then \(A^{k+1}=A^k A = 3^{k-1}(3A)=3^{k}A\). Hence holds.
🔷 Concept 2: Multiplication of Matrices and Non-Commutativity

Matrix multiplication is a row-by-column operation.

📐 1. Conformability Requirement

The product \(AB\) is defined if columns in A = rows in B. If \(A\) is \(m \times n\) and \(B\) is \(n \times p\), then \(C = AB\) is \(m \times p\).

\(A_{m \times \mathbf{n}}\)
⬇️ rows
✕
\(B_{\mathbf{n} \times p}\)
⬅️ columns
=
\(C_{m \times p}\)
result
✔ Columns of \(A\) = Rows of \(B\) → multiplication possible
⚠️ 2. Non-Commutativity

In general, \(AB \neq BA\). Even if both products are defined, equality rarely holds.

💡 Special case: If \(AB = BA\), matrices commute. If \(AB = -BA\), they anti-commute.
🧮 Solved Examples (Concept 2)
Example 1:

\(A = \begin{bmatrix} 1 & -2 & 3 \\ -4 & 2 & 5 \end{bmatrix}\), \(B = \begin{bmatrix} 2 & 3 \\ 4 & 5 \\ 2 & 1 \end{bmatrix}\). Find \(AB\).

Solution: \(AB = \begin{bmatrix} 0 & -4 \\ 10 & 3 \end{bmatrix}\).
Example 2:

\(A = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\), \(B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\). Show \(AB \neq BA\).

Solution: \(AB = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\), \(BA = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}\) → not equal.
Example 3:

\(A = \begin{bmatrix} 0 & -1 \\ 0 & 2 \end{bmatrix}\), \(B = \begin{bmatrix} 3 & 5 \\ 0 & 0 \end{bmatrix}\). Compute \(AB\).

Solution: \(AB = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}\). Conclusion: product of non-zero matrices can be zero.
Example 4 (Induction):

If \(A = \begin{bmatrix} a & b \\ 0 & 1 \end{bmatrix}\), prove \(A^n = \begin{bmatrix} a^n & \frac{b(a^n-1)}{a-1} \\ 0 & 1 \end{bmatrix}\) for \(a \neq 1\).

Proof sketch: Base \(n=1\) holds, induction step uses multiplication and simplification.
Example 5:

Find \(x\) if \(\begin{bmatrix} 1 & x & 1 \end{bmatrix} \begin{bmatrix} 1 & 3 & 2 \\ 2 & 5 & 1 \\ 15 & 3 & 2 \end{bmatrix} \begin{bmatrix} 1 \\ 2 \\ x \end{bmatrix} = 0\).

Solution: Multiplying yields \(x^2 + 16x + 28 = 0\) ⇒ \(x = -2\) or \(x = -14\).
🏆 Self Exercise (HOTS)
  1. If \(A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}\), prove \(A^n = \begin{bmatrix} 1+2n & -4n \\ n & 1-2n \end{bmatrix}\).
  2. Find \(2\times 2\) matrix \(B\) such that \(\begin{bmatrix} 1 & -2 \\ 1 & 4 \end{bmatrix}B = \begin{bmatrix} 6 & 0 \\ 0 & 6 \end{bmatrix}\).
  3. If \(A = \begin{bmatrix} \alpha & 0 \\ 1 & 1 \end{bmatrix}\) and \(B = \begin{bmatrix} 1 & 0 \\ 5 & 1 \end{bmatrix}\), find \(\alpha\) for which \(A^2 = B\).
  4. Show matrix multiplication distributes over addition: \(A(B+C)=AB+AC\).
  5. If \(AB = A\) and \(BA = B\), prove \(A^2 = A\) and \(B^2 = B\).
  6. Construct a non-zero \(3\times3\) matrix \(A\) such that \(A^2 = O\).
  7. If \(A = \begin{bmatrix} 0 & 3 \\ 4 & 7 \end{bmatrix}\) and \(f(x)=x^2-7x-12\), find \(f(A)\).
  8. If \(A^2 = I\), simplify \((A-I)^3 + (A+I)^3 – 7A\).
  9. Let \(A = \begin{bmatrix} \cos\frac{2\pi}{3} & -\sin\frac{2\pi}{3} \\ \sin\frac{2\pi}{3} & \cos\frac{2\pi}{3} \end{bmatrix}\). Find \(A^3\).
  10. Prove if \(AB=BA\), then \((A+B)(A-B)=A^2-B^2\). Does it hold if \(AB\neq BA\)?
🔥 Additional 3×3 HOTS Challenges
  1. Let \(A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & 2 \\ 2 & 2 & 1 \end{bmatrix}\). Show \(A^2 – 4A – 5I = O\) and evaluate \(A^3 – 4A^2 – 5A + A^T\).
  2. Find all \(3\times3\) matrices \(B\) commuting with \(A = \operatorname{diag}(1,2,3)\).
  3. Nilpotent matrix: \(A = \begin{bmatrix} 0 & a & b \\ 0 & 0 & c \\ 0 & 0 & 0 \end{bmatrix}\). Find \(A^2, A^3\) and conditions for index 3.
  4. Solve for \(X,Y\): \(2X+3Y = \begin{bmatrix} 2 & 3 & 5 \\ 1 & 0 & 1 \\ 2 & 1 & 3 \end{bmatrix}\), \(3X+2Y = \begin{bmatrix} 3 & 2 & 0 \\ 4 & 5 & 9 \\ 3 & 4 & 2 \end{bmatrix}\).
  5. Prove \(Tr(AB)=Tr(BA)\) and hence show no \(3\times3\) matrices satisfy \(AB-BA=I\).
  6. If \(A = \begin{bmatrix} 2 & -2 & -4 \\ -1 & 3 & 4 \\ 1 & -2 & -3 \end{bmatrix}\), verify \(A^2=A\) and find \((I+A)^{10}\).
🎯 Row × Column = Entry \(c_{ij}\)
\( \text{Row}_i(A) \)
⨯
\( \text{Column}_j(B) \)
→
\( c_{ij} \)
💡 The \((i,j)\)-entry of \(AB\) = dot product of \(i\)-th row of \(A\) and \(j\)-th column of \(B\).
📐 Master matrices: operations, properties & non-commutativity | Class XII Mathematics

Matrices Tutorial FAQs | CUET Maths Preparation

📚 Frequently Asked Questions (FAQs) – Free Matrices Tutorial Class 12

What is matrix addition and what are its important properties for CUET exam?

Matrix addition is defined only for matrices of the same order. If A = [aᵢⱼ] and B = [bᵢⱼ] are both m×n, then A+B = [aᵢⱼ + bᵢⱼ]. Key properties: Commutative Law (A+B = B+A), Associative Law [(A+B)+C = A+(B+C)], and Additive Identity (A+O = A, where O is zero matrix). For CUET Core Maths Mock Test, mastering these basics helps in solving complex problems quickly. 📘

How does scalar multiplication of matrices work? Give examples from board exams.

Scalar multiplication means multiplying every element of a matrix by a scalar (real number). If A = [aᵢⱼ] and k ∈ ℝ, then kA = [k·aᵢⱼ]. Distributive laws: k(A+B) = kA+kB and (k+l)A = kA+lA. In board exam style, you often see problems like: “If A = diag(1, –1, 2) and B = diag(2,3,–1), find 3A – B” → 3A = diag(3,–3,6) and result diag(1,–6,7). Such questions boost accuracy in CUET Core Maths Mock Test Free Online Test.

What are the conformability conditions for matrix multiplication? Why is AB ≠ BA in general?

For matrix product AB to be defined, the number of columns in A must equal the number of rows in B. If A is m×n and B is n×p, then AB is m×p. Matrix multiplication is not commutative — generally AB ≠ BA. For example, take A = [[1,0],[0,-1]] and B = [[0,1],[1,0]] ⇒ AB = [[0,1],[-1,0]] but BA = [[0,-1],[1,0]]; they differ. Understanding non-commutativity is crucial for CUET Core Maths Chapter Wise practice and advanced matrix equations.

How to solve matrix equations using addition and scalar multiplication? (Board level)

You solve matrix equations by comparing corresponding entries. Example: If 2[[x,5],[7,y-3]] + [[3,-4],[1,2]] = [[7,6],[15,14]], compute left: [[2x+3, 6],[15, 2y-4]] = [[7,6],[15,14]] ⇒ 2x+3=7 ⇒ x=2; 2y-4=14 ⇒ y=9. Our matrices tutorial covers such problems thoroughly. For CUET aspirants, practising these builds confidence in CUET Core Maths Mock Test with Answers Detailed Solutions sections. 🧠

What is the zero product property in matrices? Can product of non-zero matrices be zero?

Yes! Unlike real numbers, two non-zero matrices can multiply to give a zero matrix. Example: A = [[0,–1],[0,2]] and B = [[3,5],[0,0]] gives AB = [[0,0],[0,0]] = O. This is called nilpotent or zero divisor behaviour. Such questions frequently appear in CUET Core Mathematics Mock Test Latest Exam Pattern to test conceptual depth. Our free online test resources include many such tricky problems.

How to prove matrix power formulas using mathematical induction? (e.g., A^n = 3ⁿ⁻¹ A)

Induction is key for matrix powers. Example: A = [[1,1,1],[1,1,1],[1,1,1]] ⇒ A² = 3A. Assume Aᵏ = 3ᵏ⁻¹A, then Aᵏ⁺¹ = Aᵏ·A = 3ᵏ⁻¹A·A = 3ᵏ⁻¹(3A) = 3ᵏA. Hence proven. Similarly, if A = [[a,b],[0,1]], Aⁿ = [[aⁿ, b(aⁿ–1)/(a–1)],[0,1]]. Master these for Core Maths CUET Mock Test Free Chapter Wise to handle high-order thinking problems with ease.

What are the properties of matrix multiplication over addition? (Distributive Law)

Matrix multiplication distributes over addition: A(B+C) = AB + AC and (A+B)C = AC + BC, provided conformability holds. This is used extensively in simplifying matrix expressions. For example, (A+I)(A–I) = A² – I if A and I commute. For CUET preparation, our detailed solutions show step-by-step distribution. Visit our website for CUET Core Maths Mock Test Free Online Test that tests distributive properties in 2×2 and 3×3 matrices.

How to find unknown matrices X and Y from system of matrix equations? (Board 3×3)

Given 2X+3Y = P and 3X+2Y = Q, solve like linear equations: multiply first by 2, second by 3 and subtract, or add/subtract. Example: 2X+3Y = [[2,3,5],[1,0,1],[2,1,3]] and 3X+2Y = [[3,2,0],[4,5,9],[3,4,2]]. Solve: Adding gives 5(X+Y) = … subtract gives X–Y = … then find X and Y. Such problems appear in Core Maths CUET Mock Test with Answers Detailed Solutions to strengthen algebraic manipulation in matrices. 📊

What are nilpotent matrices and how do you find A², A³ for upper triangular nilpotent?

A nilpotent matrix is a square matrix such that Aᵏ = O for some positive integer k. For strict upper triangular 3×3: A = [[0,a,b],[0,0,c],[0,0,0]] ⇒ A² = [[0,0,ac],[0,0,0],[0,0,0]] and A³ = O. This is index 3 nilpotent if a,c ≠ 0. Such concepts are vital for CUET advanced section and JEE foundation. Our Matrices Tutorial page offers interactive examples on nilpotent & idempotent matrices with free practice tests.

How can CUET aspirants use matrix operations mock tests to improve accuracy?

Regular practice with CUET Core Maths Mock Test Free Chapter Wise helps you master matrix addition, scalar multiplication, and non-commutative products. Our platform provides CUET Core Mathematics Mock Test Latest Exam Pattern with detailed step-by-step solutions, so you can track mistakes in conformability, sign errors, or power expansion. Start with our matrices topic tests, then attempt full-length mock tests to boost speed and conceptual clarity. 📈

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