Class 12 Maths Chapter 3: Matrices – Concepts, NCERT Solutions & CBSE PYQs Explained

 Here’s the comprehensive guide for Chapter 3: Matrices (Class 12 CBSE Maths):

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🧠 A. Key Concepts & Properties

1. Matrix Definition & Order
   A matrix is a rectangular array $[a_{ij}]$ of order $m \times n$ containing real numbers/functions.

2. Types of Matrices :

   • Row, Column, Square, Rectangular

   • Zero, Identity, Diagonal, Scalar

   • Symmetric ($A^T = A$), Skew-symmetric ($A^T = -A$), Transpose ($A^T$)

3. Matrix Operations:

   • Sum/Subtraction: Element-wise, requires same order 

   • Scalar Multiplication

   • Matrix Multiplication: Product defined if columns of A = rows of B; not commutative .

4. Elementary Row/Column Operations & Inverse:

   • Inverse ($A^{-1}$) exists only for square matrices; findable via elementary operations 

5. Properties of Transpose:

   • $(A^T)^T = A$, $(AB)^T = B^T A^T$ 

6. Symmetric/Skew Decomposition:

   • Any square $A$ can be written as $S + K$, where $S = (A + A^T)/2$ and $K = (A - A^T)/2$ 

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📘 B. NCERT Exercise Breakdown & Solutions

(NCERT Chapter 3 exercises generally include)

• Finding transpose, checking equality

• Sum, scalar multiplication, and product of matrices

• Finding unknown elements/matrices from equations

• Determining inverse using elementary operations

• Symmetric/skew decomposition

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📝 C. CBSE Previous Year Questions with Model Solutions

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✅ PYQ 2019 (Delhi):

Q. If $3A - B = \begin{pmatrix}5&0\\1&1\end{pmatrix}$ and $B = \begin{pmatrix}4&3\\2&5\end{pmatrix}$, find A. 
Ans:

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✅ PYQ 2018 (CBSE Syllabus):

Q. If $A$ is skew-symmetric, find a and b for $A = \begin{pmatrix}0 & a & -3\\2&0&-1\\b&1&0\end{pmatrix}$.
Ans:
Skew-symmetric ⇒ $A^T = -A$. Equating gives $a+2=0 \Rightarrow a=-2$; $b=-1$.

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✅ PYQ 2016 (All India):

Q. Given $A = P + Q$, where P is symmetric and Q is skew-symmetric, find P if $A = \begin{pmatrix}3&5\\7&9\end{pmatrix}$.
Ans:

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✅ PYQ 2023 (MCQ style):

Q. If $A=[a_{ij}]$ skew-symmetric, then $|A| = 0$ for odd order matrices.
Ans: True — determinant of odd-order skew-symmetric is always zero .

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🔍 D. Exam Strategy & Tips

• Use step-by-step matrix equations clearly.

• Label and verify dimensions before operations.

• For inverse, augment with identity and apply row reductions.

• For symmetric/skew decomposition, always compute $(A ± A^T)/2$.

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#Tags:

#Class12Maths, #Matrices, #MatrixOperations, #NCERTSolutions, #CBSEPYQs, #InvertibleMatrix, #SymmetricMatrix, #SkewSymmet