Class 12 Maths Matrices – NCERT Formulas & Examples

Class 12 Maths Chapter 3: Matrices

Welcome to FUZY MATH ACADEMY. This chapter covers Matrices including types, operations, determinants, inverses, rank, and applications. We provide 25 NCERT + IIT-level questions and 15 FAQs, all solved step by step with our 24/7 AI Math Solver and LMS.

Formulas Used in Matrices

• Addition: (A + B) = [a_ij + b_ij]
• Scalar Multiplication: kA = [k × a_ij]
• Matrix Multiplication: (AB)_ij = Σ a_ik b_kj
• Transpose: (A^T)_ij = a_ji
• Determinant: 2×2: |A| = ad - bc, 3×3: |A| = a(ei − fh) − b(di − fg) + c(dh − eg)
• Inverse: A⁻¹ = 1/|A| × adj(A), if |A| ≠ 0

Examples

Example 1: Matrix Addition

Let A=[[1,2],[3,4]] and B=[[2,0],[1,2]]. Find A+B.

Solution: A+B=[[3,2],[4,6]]

Example 2: Matrix Multiplication

Let A=[[1,2],[0,1]] and B=[[2,1],[3,0]]. Find AB.

Solution: AB=[[8,1],[3,0]]

25 NCERT + IIT-Level Questions

Question 1

Find the sum of A=[[1,2],[3,4]] and B=[[2,0],[1,2]].

A+B=[[3,2],[4,6]]

Question 2

Find the product of A=[[1,2],[0,1]] and B=[[2,1],[3,0]].

AB=[[8,1],[3,0]]

Question 3

Find the transpose of A=[[1,2,3],[4,5,6]].

A^T=[[1,4],[2,5],[3,6]]

Question 4

Find the determinant of A=[[1,2],[3,4]].

|A|=-2

Question 5

Find the inverse of A=[[2,1],[1,1]].

A^-1=[[1,-1],[-1,2]]

Question 6

Verify (AB)^T=B^TA^T for A=[[1,2],[0,1]], B=[[2,0],[1,3]].

AB=[[4,6],[1,3]], (AB)^T=[[4,1],[6,3]], B^TA^T=[[4,1],[6,3]] ✅

Question 7

Check if A=[[1,2],[2,4]] is invertible.

|A|=0 ⇒ Not invertible

Question 8

Find k such that A=[[2,k],[3,4]] is singular.

|A|=2*4 - 3*k = 0 ⇒ k=8/3

Question 9

If A=[[1,2],[3,4]], B=[[0,1],[1,0]], find A+B and AB.

A+B=[[1,3],[4,4]], AB=[[2,1],[4,3]]

Question 10

Find determinant of A=[[1,2,3],[0,1,4],[5,6,0]].

|A|=1(1*0-4*6)-2(0*0-4*5)+3(0*6-1*5)=1

Question 11

Find A² if A=[[1,2],[3,4]].

A²=[[7,10],[15,22]]

Question 12

Find adjoint of A=[[1,2],[3,4]].

adj(A)=[[4,-2],[-3,1]]

Question 13

Find A^-1 for A=[[1,2],[3,4]].

|A|=-2, adj(A)=[[4,-2],[-3,1]] ⇒ A^-1[[-2,1],[3/2,-1/2]]

Question 14

Verify A×A^-1=I for A=[[2,1],[1,1]].

A×A^-1=I

Question 15

Find X if AX=I for A=[[1,2],[3,4]].

X=A^-1

Question 16

Find the rank of A=[[1,2,3],[4,5,6],[7,8,9]].

Rank=2

Question 17

If A=[[1,0,2],[2,1,3],[1,0,2]], find its rank.

Rank=2

Question 18

Verify A(B+C)=AB+AC for A=[[1,0],[0,1]], B=[[2,1],[1,2]], C=[[0,1],[1,0]].

AB+AC=[[2,2],[2,2]] = A(B+C)

Question 19

Find X if AX=[[5,11],[11,25]] for A=[[1,2],[3,4]].

X=A

Question 20

Compute determinant of A=[[2,1,3],[1,0,2],[3,4,5]].

|A|=2(0*5-2*4)-1(1*5-2*3)+3(1*4-0*3)=-16

Question 21

Find inverse of A=[[1,2,3],[0,1,4],[5,6,0]].

A^-1=calculated using adj(A)/|A|

Question 22

Find X such that AX=B where A=[[1,1],[1,2]], B=[[3,5]].

X=A^-1B=[[1,2]]

Question 23

Find if matrix A=[[1,2],[2,3]] is symmetric.

Yes, A=A^T

Question 24

Check if A=[[1,2],[3,4]] is skew-symmetric.

No, A≠-A^T

Question 25

Find the rank of A=[[1,2,3],[4,5,6],[7,8,9]].

Rank=2

15 FAQs

FAQ 1: What is a matrix?

A matrix is a rectangular array of numbers arranged in rows and columns.

FAQ 2: What is the order of a matrix?

Order of a matrix = m×n (rows × columns)

FAQ 3: What is determinant?

Determinant is a scalar value representing certain properties like invertibility of square matrices.

FAQ 4: How to find inverse of a matrix?

A^-1 = adj(A)/|A| if |A|≠0

FAQ 5: What is rank?

Rank is the maximum number of linearly independent rows or columns.

FAQ 6: What is singular matrix?

A matrix with determinant 0 is singular.

FAQ 7: What is non-singular matrix?

A matrix with non-zero determinant is non-singular.

FAQ 8: What is symmetric matrix?

A matrix A is symmetric if A=A^T

FAQ 9: What is skew-symmetric matrix?

A matrix A is skew-symmetric if A=-A^T

FAQ 10: What is transpose of a matrix?

Transpose of A, denoted A^T, is obtained by interchanging rows and columns.

FAQ 11: What is identity matrix?

A square matrix with 1's on the diagonal and 0 elsewhere.

FAQ 12: How to multiply matrices?

AB is defined if columns of A = rows of B. (AB)_ij=Σ a_ik b_kj

FAQ 13: How to add matrices?

FUZY Math Academy – Online Math Tuition for Classes 5 to 12

Personalized Online Coaching | AI-Powered LMS | 24/7 AI Math Solver. Learn from expert teachers and get step-by-step NCERT solutions for Classes 5–12.

Read Full Post

Visit www.fuzymathacademy.com for more courses and free demo classes.