Class 12 Maths · Chapter 3
Unit: Algebra
Tentative Study Share: 5.0 Marks
Chapter 3: Matrices — NCERT Solutions, Notes, Formula Sheet & PYQs
Complete NCERT exercise solutions, revision notes, formula cards, printable mind map, and solved board questions for Class 12 Matrices.
NCERT Exercises in Chapter 3: Matrices
📘 NCERT Concept Notes — Matrices
01 · Read the shape before the entries
A matrix is an ordered rectangular array. Row index comes first. Equal matrices must have the same shape and equal corresponding entries. A scalar or identity matrix is square, while a zero matrix can be rectangular.
02 · Addition follows entries; multiplication follows rows and columns
Addition is entrywise and needs equal orders. In AB, the column count of A must match the row count of B. Matrix multiplication generally depends on order.
03 · Transpose reverses a product
Transpose exchanges rows and columns. Symmetry compares a matrix with its transpose. Any real square matrix splits naturally into symmetric and skew-symmetric parts.
04 · Inverse means undoing on both sides
An inverse is a matrix, not a collection of entrywise reciprocals. Keep the order of factors when undoing a product. An inverse, if it exists, is unique.
⚡ Key Formulas & Results — Matrices
Order and equality
What does aᵢⱼ mean, and when can two matrices be equal?
\(A=[a_{ij}]_{m\times n},\quad A=B\iff a_{ij}=b_{ij}\ \forall i,j\)
Diagonal, scalar and identity
How do diagonal, scalar and identity matrices differ?
\(D=\operatorname{diag}(d_1,\ldots,d_n),\quad kI=\operatorname{diag}(k,\ldots,k)\)
Addition and scaling
What is the entry rule for αA+βB?
\((\alpha A+\beta B)_{ij}=\alpha a_{ij}+\beta b_{ij}\)
Matrix multiplication
What are the shape and entries of AB?
\(A_{m\times n}B_{n\times p}=C_{m\times p},\quad c_{ij}=\sum_{k=1}^n a_{ik}b_{kj}\)
Product laws and traps
Which familiar algebra laws remain valid for matrix products?
\(A(BC)=(AB)C,\quad A(B+C)=AB+AC\)
Transpose laws
What happens to the order of factors under transpose?
\((A^T)^T=A,\quad(A+B)^T=A^T+B^T,\quad(AB)^T=B^TA^T\)
Symmetric and skew symmetric
What are the defining transpose equations?
\(S^T=S,\quad K^T=-K\)
Derive the decomposition
How can a square matrix be split without solving for unknown entries?
\(A=\frac{A+A^T}{2}+\frac{A-A^T}{2}\)
Inverse matrix
What identities define the inverse of A?
\(AA^{-1}=A^{-1}A=I\)
Why the inverse is unique
If B and C are both inverses, how do you prove they are equal?
\(B=B(AC)=(BA)C=C\)
🎯 CBSE 2026 Board PYQs from Matrices
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q2
Which of the following cannot be the order of a row-matrix?
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q3
Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)′ = A′ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q6
If A and B are skew symmetric matrices of same order, then which of the following matrices is also skew…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/1 · Q33
If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A⁻¹ and use it to solve the system: −2y + z = 7, 2x − y − z…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q13
Which cannot be the order of a column matrix?
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q14
Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)′ = A′ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q17
For symmetric matrices A,B of the same order, which expression is always skew-symmetric?
CBSE Class 12 Mathematics 2026 Main Paper 65/1/2 · Q35
If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A⁻¹ and use it to solve the system: −2y + z = 7, 2x − y − z…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q6
Which property must hold for every scalar matrix of order 3?
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q7
Which of the following properties is/are true for two matrices of suitable orders? (i) (A + B)′ = A′ +…
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q10
For symmetric matrices A and B of equal order, classify AB−BA.
CBSE Class 12 Mathematics 2026 Main Paper 65/1/3 · Q34
If A = [[0,2,1],[-2,-1,-2],[1,-1,0]], find A⁻¹ and use it to solve the system: −2y + z = 7, 2x − y − z…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q2
For same-order square matrices A and B, which statements always hold? (i) (A+B)(A−B)=A²−B². (ii) AB=BA…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q3
If [[1,a,b],[−1,2,c],[0,5,3]] is symmetric, find 3a+b+c.
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q4
For A=[[cos x,−sin x],[sin x,cos x]], solve A+Aᵀ=I for x in [0,π/2].
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q5
For a square matrix A, which expression equals (3A)⁻¹ when it exists?
CBSE Class 12 Mathematics 2026 Main Paper 65/2/1 · Q32
For P=[[1,−1,0],[2,3,4],[0,1,2]] and Q=[[2,2,−4],[−4,2,−4],[2,−1,5]], compute QP. Use matrices to solve…
CBSE Class 12 Mathematics 2026 Main Paper 65/2/2 · Q13
For same-order square matrices A and B, which statements always hold? (i) (A+B)(A−B)=A²−B². (ii) AB=BA…
📝 Chapterwise Practice Questions — Matrices (104 Questions)
Class 12 Matrices · Mcq Single Correct
Let A be a 2×2 matrix with aᵢⱼ=1 for i≠j and 0 for i=j. Find A².
Class 12 Matrices · Mcq Single Correct
Let A=\(\begin{bmatrix}\cos\alpha & -\sin\alpha\\\sin\alpha & \cos\alpha\end{bmatrix}\). Which listed…
Class 12 Matrices · Mcq Single Correct
If \(\begin{bmatrix}2a+b & a−2b\\5c−d & 4c+3d\end{bmatrix}\)=\(\begin{bmatrix}4 & -3\\11 &…
Class 12 Matrices · Mcq Single Correct
How many 3×3 matrices have every entry either 2 or 0?
Class 12 Matrices · Mcq Single Correct
Find x+y for constants x,y such that the identity A⁻¹=xA+yI holds for every real square matrix A…
Class 12 Matrices · Mcq Single Correct
If \(\begin{bmatrix}2 & 0\\5 & 4\end{bmatrix}\)=P+Q, where P is symmetric and Q skew-symmetric, find Q.
Class 12 Matrices · Mcq Single Correct
If A=\(\begin{bmatrix}3 & 4\\5 & 2\end{bmatrix}\) and 2A+B=O, find B.
Class 12 Matrices · Mcq Single Correct
If A=\(\begin{bmatrix}0 & 1\\0 & 0\end{bmatrix}\), find A²⁰²³.
Class 12 Matrices · Mcq Single Correct
Let A,B,C have orders 2×3, 3×2, 3×3 respectively. Which expression is undefined?
Class 12 Matrices · Mcq Single Correct
For real matrices A and B of the same order, which property is always satisfied by ABᵀ−BAᵀ?
Class 12 Matrices · Mcq Single Correct
A 3×3 matrix P satisfies Pᵀ=2P+I. Find P.
Class 12 Matrices · Mcq Single Correct
For A=[aᵢⱼ]₃ₓ₃, let aᵢⱼ=2i+3j if i j. How many entries exceed 5?
Class 12 Matrices · Mcq Single Correct
If A=\(\begin{bmatrix}0 & 2\\3 & -4\end{bmatrix}\) and kA=\(\begin{bmatrix}0 & 3a\\2b &…
Class 12 Matrices · Mcq Single Correct
If A²=I, simplify (A−I)³+(A+I)³−7A.
Class 12 Matrices · Mcq Single Correct
For A=\(\begin{bmatrix}4 & 2\\-1 & 1\end{bmatrix}\), find (A−2I)(A−3I).
Class 12 Matrices · Mcq Single Correct
Let A=\(\begin{bmatrix}2 & -1 & 3\\-4 & 5 & 1\end{bmatrix}\) and B=\(\begin{bmatrix}2 & 3\\4 & -2\\1 &…
Class 12 Matrices · Mcq Single Correct
If A=\(\begin{bmatrix}α & β\\γ & −α\end{bmatrix}\) satisfies A²=3I, which relation holds?
Class 12 Matrices · Mcq Single Correct
For A=\(\begin{bmatrix}0 & 1\\-1 & 0\end{bmatrix}\), if (3I+4A)(3I−4A)=x²I, find all real x.
Class 12 Matrices · Mcq Single Correct
A has order 3×n and B has order m×5. Assuming C=5A+3B is defined, what is the order of C?
Class 12 Matrices · Mcq Single Correct
For A=\(\begin{bmatrix}1 & -1 & 1\\1 & -1 & 1\\1 & -1 & 1\end{bmatrix}\), find A⁵−A⁴−A³+A².
Class 12 Matrices · Mcq Single Correct
If A²−A+I=O, express A⁻¹ in terms of A and I.
Class 12 Matrices · Mcq Single Correct
Let A=\(\begin{bmatrix}1 & 0\\2 & 1\end{bmatrix}\) and B=\(\begin{bmatrix}x & 0\\1 & 1\end{bmatrix}\)…
Class 12 Matrices · Mcq Single Correct
For A=\(\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}\) and B=\(\begin{bmatrix}1 & 1\\0 & 0\end{bmatrix}\)…
Class 12 Matrices · Mcq Single Correct
For arbitrary same-order real skew-symmetric matrices A and B, which condition is necessary and…