NCERT Solutions for Class 12 Maths Chapter 3 Matrices

Mathematics plays a key role in developing analytical and reasoning abilities, and Chapter 3: Matrices forms an essential part of Class 12 Maths. This chapter introduces students to the concept of matrices, their types, operations such as addition, subtraction, and multiplication, along with the transpose and properties of matrices. It also builds the base for advanced topics like determinants and systems of linear equations.

These NCERT Solutions for Class 12 Maths Chapter 3 include important CBSE board questions asked between 2020 and 2025. Each solution is explained in a clear, step-by-step manner, using simple mathematical reasoning to help students strengthen their concepts and score confidently in the board examinations.

NCERT Solutions for Class 12 Maths Chapter 3 Matrices

Matrices

Medium

Q.

$\begin{array}{l}\mathrm{In} \mathrm{a} \mathrm{matrix} \mathrm{A}=\left[\begin{array}{l}2\hspace{0.17em}\hspace{0.17em}5 19 -7\\ 35-2 \hspace{0.17em}\frac{5 }{2}12\\ \sqrt{3}\hspace{0.17em}\hspace{0.17em}1-5 17\end{array}\right],\hspace{0.17em}\hspace{0.17em}\mathrm{write}:\\ \left(\mathrm{i}\right) \mathrm{The} \mathrm{order} \mathrm{of} \mathrm{the} \mathrm{matrix}\\ \left(\mathrm{ii}\right) \mathrm{Then} \mathrm{umber} \mathrm{of} \mathrm{elements},\\ \left(\mathrm{iii}\right) \mathrm{Write} \mathrm{the} \mathrm{elements} {\mathrm{a}}_{13},{\mathrm{a}}_{21},{\mathrm{a}}_{33},{\mathrm{a}}_{24},{\mathrm{a}}_{23}.\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Compute} \mathrm{the} \mathrm{following}:\\ \left(\mathrm{i}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\mathrm{a} \mathrm{b}\\ -\mathrm{b} \mathrm{a}\end{array}\right]+\left[\begin{array}{l}\mathrm{a} \mathrm{b}\\ \mathrm{b} \mathrm{a}\end{array}\right]\\ \left(\mathrm{ii}\right) \left[\begin{array}{l}{\mathrm{a}}^2+{\mathrm{b}}^{2 \; }{\mathrm{b}}^2+{\mathrm{c}}^2\\ {\mathrm{a}}^2+{\mathrm{c}}^{2 \; }{\mathrm{a}}^2+{\mathrm{b}}^2\end{array}\right]\\ \left(\mathrm{iii}\right) \left[\begin{array}{l}-1 4-6\\ 8 5 16\\ 2 8\mathrm{\hspace{0.17em}}\hspace{0.17em} 5\end{array}\right]+\left[\begin{array}{l}12 7 6\\ 8 \; 0 \; 5\\ 3 2 \; 4\end{array}\right]\mathrm{\hspace{0.17em}}\\ \left(\mathrm{iv}\right) \left[\begin{array}{l}{\cos }^2\mathrm{x} {\sin }^2\mathrm{x}\\ {\sin }^2\mathrm{x} {\cos }^2\mathrm{x}\end{array}\right]+\left[\begin{array}{l}{\sin }^2\mathrm{x} {\cos }^2\mathrm{x}\\ {\cos }^2\mathrm{x} {\sin }^2\mathrm{x}\end{array}\right]\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Using} \mathrm{elementary} \mathrm{transformations}, \mathrm{find}\\ \mathrm{the} \mathrm{inverse} \mathrm{of} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{matrices}\hspace{0.33em}\left[\begin{array}{l}2\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-3 \hspace{0.33em}\hspace{0.33em}3\\ 2\hspace{0.33em} \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}2\hspace{0.33em} 3\hspace{0.33em}\\ 3\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}-2 \hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}2\hspace{0.33em}\end{array}\right]\end{array}$

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Matrices

Easy

Q.

$\begin{array}{l}\mathbf{If}\mathrm{\hspace{0.17em}}\mathbf{A}=\left[\begin{array}{l}\mathbf{cos}\mathrm{\alpha }-\mathbf{sin}\mathrm{\alpha }\\ \mathbf{sin}\mathrm{\alpha }\hspace{0.17em}\hspace{0.17em}\mathbf{cos}\mathrm{\alpha }\end{array}\right],\mathrm{ }\mathrm{thenA}+\mathrm{A}\text{'}=\mathrm{I}, \mathrm{if} \mathrm{the} \mathrm{value} \mathrm{of} \mathrm{\alpha }\mathrm{\hspace{0.17em}}\mathrm{is}\\ \left(\mathrm{A}\right) \frac{\mathrm{\pi }}{6}\\ \left(\mathrm{B}\right) \frac{\mathrm{\pi }}{3}\\ \left(\mathrm{C}\right) \mathrm{\pi }\\ \left(\mathrm{D}\right) \frac{3\mathrm{\pi }}{2}\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\left(\text{i}\right)\mathit{ }\mathit{\text{Show tha t the matrix A=}}\left[\begin{array}{l}1-15\\ 121\\ 513\end{array}\right]\mathit{ }\mathit{i}\mathit{\text{s a symmetric matrix.}}\\ \\ \left(ii\right)\mathit{\text{ Show tha tthe matrix A=}}\left[\begin{array}{l}01-1\\ -101\\ 1-10\end{array}\right]\mathit{ }\mathit{\text{is a skew}}\\ \mathit{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}symmetric matrix}}\mathit{.}\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{For} \mathrm{the} \mathrm{matrices} \mathrm{A} \mathrm{and} \mathrm{B},\mathrm{verify} \mathrm{that} \left(\mathrm{AB}\right)\text{'}=\mathrm{B}\text{'}\mathrm{A}\text{'},\hspace{0.17em} \mathrm{where}\\ \left(\mathrm{i}\right) \mathrm{A}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}1\\ -4\\ \hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}3\end{array}\right],\hspace{0.17em}\hspace{0.17em}\mathrm{B}=[-121]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\\ \left(\mathrm{ii}\right) \mathrm{A}=\left[\begin{array}{l}0\\ 1\\ 2\end{array}\right],\mathrm{\hspace{0.17em}}\mathrm{B}=\left[157\right]\end{array}$

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Matrices

Medium

Q.

$\text{If\hspace{0.17em} A=}\text{[}\begin{array}{l}\frac{2}{3}1\frac{5}{3}\\ \frac{1}{3}\frac{2}{3}\frac{4}{3 }\\ \frac{7}{3}2\frac{2}{3}\end{array}] \& \mathrm{B}=\left[\begin{array}{l}\frac{2}{5}\frac{3}{5}1\\ \frac{1}{5}\frac{2}{5}\frac{4}{5}\\ \frac{7}{5}\frac{6}{5}\frac{2}{5}\end{array}\right], \mathrm{then} \mathrm{compute} 3\mathrm{A}-5\mathrm{B}.$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{If} \mathrm{A}= \left[\begin{array}{l}12 -3\\ 50\mathrm{\hspace{0.17em}}\hspace{0.17em} \hspace{0.17em}2\\ 1\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} -1\hspace{0.17em}\hspace{0.17em}1\end{array}\right],\mathrm{\hspace{0.17em}}\mathrm{B}=\left[\begin{array}{l}3 -12\\ 4\mathrm{\hspace{0.17em}}\hspace{0.17em} \hspace{0.17em}25\\ 2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 03\end{array}\right] \mathrm{and} \mathrm{C}=\left[\begin{array}{l}4 12\\ 0 32\\ 1 -23\end{array}\right],\\ \mathrm{then} \mathrm{compute} (\mathrm{A}+\mathrm{B}) \mathrm{and} (\mathrm{B}-\mathrm{C}). \mathrm{\hspace{0.17em}}\mathrm{Also}, \mathrm{verify} \mathrm{that}\\ \mathrm{A}+(\mathrm{B}-\mathrm{C}) = (\mathrm{A}+\mathrm{B})-\mathrm{C}.\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Compute} \mathrm{the} \mathrm{indicated} \mathrm{products}:\\ \left(\mathrm{i}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\mathrm{a} \mathrm{b}\\ -\mathrm{b} \mathrm{a}\end{array}\right] \left[\begin{array}{l}\mathrm{a}-\mathrm{b}\\ \mathrm{b}\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}\mathrm{a}\end{array}\right]\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{ii}\right) \left[\begin{array}{l}1\\ 2\\ 3\end{array}\right] \left[234\right]\\ \left(\mathrm{iii}\right) \left[\begin{array}{l}1 -2\\ 2\hspace{0.17em} \hspace{0.17em}\mathrm{\hspace{0.17em}}3\end{array}\right] \left[\begin{array}{l}1 2 3\\ 2 3\mathrm{ }1\end{array}\right]\\ \left(\mathrm{iv}\right) \left[\begin{array}{l}2 3 4\\ 3 4 5\\ 4 5 6\end{array}\right] \left[\begin{array}{l}1\mathrm{ }-3 5\\ 0\hspace{0.17em} \hspace{0.17em}\mathrm{\hspace{0.17em}}2 4\\ 3\hspace{0.17em}\hspace{0.17em}\hspace{0.17em} 0 5\end{array}\right]\end{array}$

$\begin{array}{l}\left(v\right)\left[\begin{array}{l}21\\ \mathrm{3\; }2\\ -\mathrm{1\; }1\end{array}\right]\left[\begin{array}{l}\mathrm{1\; }\mathrm{0\; }1\\ -\mathrm{1 }21\end{array}\right]\\ \left(vi\right)\left[\begin{array}{l}3-\mathrm{1\; }3\\ -1\mathrm{0\; }2\end{array}\right]\left[\begin{array}{l}2-3\\ 1\mathrm{ \; 0}\\ 3\mathrm{ \; 1}\end{array}\right]\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{In} \mathrm{the} \mathrm{matrix} \mathrm{A}= \left[\begin{array}{l}2 4\\ 3 2\end{array}\right],\mathrm{\hspace{0.17em}}\mathrm{B}=\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}1 3\\ -2 5\end{array}\right],\mathrm{\hspace{0.17em}}\mathrm{C}=\left[\begin{array}{l}-2 5\\ \hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}3 \; 4\end{array}\right]\\ \mathrm{Find} \mathrm{each} \mathrm{of} \mathrm{the} \mathrm{following}:\\ \left(\mathrm{i}\right) \mathrm{A}+\mathrm{B} \\ \left(\mathrm{ii}\right) \mathrm{A}-\mathrm{B} \\ \left(\mathrm{iii}\right) 3\mathrm{A}-\mathrm{C} \\ \left(\mathrm{iv}\right) \mathrm{AB}\left(\mathrm{v}\right) \mathrm{BA}\end{array}$

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Matrices

Easy

Q.

If a matrix has 24 elements, what are the  possible orders it can have? What, if it has 13 elements?

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Matrices

Easy

Q.

The number of all possible matrices of order 3 × 3 with each entry 0 or 1 is:

(A) 27    

(B) 18    

(C) 81   

(D) 512

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Matrices

Easy

Q.

$\begin{array}{l}\mathrm{Which} \mathrm{of} \mathrm{the} \mathrm{given} \mathrm{values} \mathrm{of} \mathrm{x} \mathrm{and} \mathrm{y} \mathrm{make} \mathrm{the} \mathrm{following}\\ \mathrm{pair} \mathrm{of} \mathrm{matrice} \mathrm{sequal}\\ \left[\begin{array}{l}3\mathrm{x}+7 \; 5\\ \mathrm{y}+1 2-3\mathrm{x}\end{array}\right],\mathrm{\hspace{0.17em}}\left[\begin{array}{l}0 \mathrm{y}-2\\ 8 \; 4\end{array}\right]\\ \left(\mathrm{A}\right) \mathrm{x}=-\frac{1}{3},\hspace{0.17em}\hspace{0.17em}\mathrm{y}=7\left(\mathrm{B}\right) \mathrm{Not} \mathrm{possiblet} \mathrm{of} \mathrm{ind}\\ \left(\mathrm{C}\right) \mathrm{y}=7,\mathrm{x}=\frac{-2}{3}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(\mathrm{D}\right)\mathrm{x}=-\frac{1}{3},\mathrm{\hspace{0.17em}}\mathrm{y}=\frac{-2}{3}\end{array}$

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Matrices

Easy

Q.

$$\begin{gathered} \begin{array}{l}\mathrm{A}={ \left[{\mathrm{a}}_{\mathrm{ij}}\right]}_{\mathrm{m}\times \mathrm{n}}\mathrm{is} \mathrm{a} \mathrm{square} \mathrm{matrix}, \mathrm{if}\\ \left(\mathrm{A}\right) \mathrm{m}<\mathrm{n}\\ \left(\mathrm{B}\right) \mathrm{m}>\mathrm{n}\\ \left(\mathrm{C}\right) \mathrm{m}=\mathrm{n} \\ \left(\mathrm{D}\right) \mathrm{None} \mathrm{of} \mathrm{these}\end{array} \end{gathered}$$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Find} \mathrm{the} \mathrm{value} \mathrm{of}\mathrm{\hspace{0.17em}}\mathrm{a}, \mathrm{b}, \mathrm{c}\hspace{0.17em} \mathrm{and} \mathrm{\hspace{0.17em}}\mathrm{d} \mathrm{from} \mathrm{thee} \mathrm{quation}:\\ \left[\begin{array}{l}\mathrm{a}-\mathrm{b} 2\mathrm{a}+\mathrm{c}\\ 2\mathrm{a}-\mathrm{b} 3\mathrm{c}+\mathrm{d}\end{array}\right]=\left[\begin{array}{l}-1 \hspace{0.17em}\hspace{0.17em}5\\ \hspace{0.17em}\hspace{0.17em}0 13\end{array}\right]\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Find} \mathrm{the} \mathrm{values} \mathrm{of} \mathrm{x}, \mathrm{y} \mathrm{and} \mathrm{z} \mathrm{from} \mathrm{the} \mathrm{following}\hspace{0.17em} \mathrm{equations}:\\ \left(\mathrm{i}\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{l}4 \; 3\\ \mathrm{x} 5\end{array}\right]=\left[\begin{array}{l}\mathrm{y} \mathrm{z}\\ 1 5\end{array}\right]\hspace{0.17em}\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\left(\mathrm{ii}\right) \left[\begin{array}{l}\mathrm{x}+\mathrm{y} 3\\ 5+\mathrm{z} \mathrm{xy}\end{array}\right]=\left[\begin{array}{l}6 2\\ 5 8\end{array}\right]\\ \left(\mathrm{iii}\right) \left[\begin{array}{l}\mathrm{x}+\mathrm{y}+\mathrm{z}\\ \mathrm{x}+\mathrm{z}\\ \mathrm{y}+\mathrm{z}\end{array}\right]=\left[\begin{array}{l}9\\ 5\\ 7\end{array}\right]\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathbf{Construct}\mathrm{ }\mathbf{a}\mathrm{ }\mathbf{3}\mathrm{ }\times \mathrm{ }\mathbf{4}\mathrm{ }\mathbf{matrix},\mathrm{ }\mathbf{whose}\mathrm{ }\mathbf{elements}\mathrm{ }\mathbf{are}\hspace{0.17em}\hspace{0.17em}\mathbf{given}\mathrm{ }\mathbf{by}:\\ \left(\mathrm{i}\right) \mathrm{\hspace{0.17em}}{\mathrm{a}}_{\mathrm{ij}}=\frac{1}{2}|-3\mathrm{i}+\mathrm{j}| \left(\mathrm{ii}\right) {\mathrm{a}}_{\mathrm{ij}}=2\mathrm{i}-\mathrm{j}\end{array}$

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Matrices

Medium

Q.

$\begin{array}{l}\mathrm{Constructa} 2\times 2\mathrm{matrix}, \mathrm{A}= \left[{\mathrm{a}}_{\mathrm{ij}}\right],\mathrm{whose} \mathrm{ }\mathrm{elementsare}\\ \mathrm{givenby}:\mathrm{\hspace{0.17em}} \left(\mathrm{i}\right)\mathrm{ }{\mathrm{a}}_{\mathrm{ij}}=\frac{{(\mathrm{i}+\mathrm{j})}^2}{2} \left(\mathrm{ii}\right) {\mathrm{a}}_{\mathrm{ij}}=\frac{\mathrm{i}}{\mathrm{j}} \left(\mathrm{iii}\right) {\mathrm{a}}_{\mathrm{ij}}=\frac{{(\mathrm{i}+2\mathrm{j})}^2}{2}\end{array}$

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Matrices

Easy

Q.

If a matrix has 18 elements, what are the  possible orders it can have? What, if it has 5 elements?

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Matrices

Medium

Q.

$\text{If A =}\left[\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}31\\ -12\end{array}\right], \mathrm{\hspace{0.17em}}\mathrm{show}\hspace{0.17em}\hspace{0.17em}\mathrm{that} {\mathrm{A}}^2-5\mathrm{A}+7\mathrm{I}=0.$

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Subject: Mathematics  |  Chapter: Matrices  |  Difficulty Level: Medium

Q1. (i) Show that the matrix

A =

1    -15
1    21
5    13

is a symmetric matrix.

(ii) Show that the matrix

A =

0    1    -1
-1    0    1
1    -1    0

is a skew-symmetric matrix.

A. (i) To show A is symmetric:

A is symmetric if A^T = A, i.e., a_ij = a_ji for all i, j.

Given

A =
1    -15
1    21
5    13

But the matrix has order 3 × 2, so A^T will be 2 × 3.
Since the order changes, A^T cannot be equal to A.
Therefore, this matrix cannot be symmetric.

Conclusion: The given matrix in (i) is not symmetric (because it is not a square matrix).

(ii) To show A is skew-symmetric:

A is skew-symmetric if A^T = −A, i.e., a_ji = −a_ij for all i, j and all diagonal entries are 0.

Given

A =
0    1    -1
-1    0    1
1    -1    0

Now the transpose is:

A^T =
0    -1    1
1    0    -1
-1    1    0

Also,

−A =
0    -1    1
1    0    -1
-1    1    0

Since A^T = −A, the given matrix is skew-symmetric.

Subject: Mathematics  |  Chapter: Matrices  |  Difficulty Level: Medium

Q2. If A =

2    3
1    4

Find |A| and state whether A is invertible or not.

A1.

For a 2×2 matrix, if

A =
a    b
c    d

then |A| = ad − bc.

Here, a = 2, b = 3, c = 1, d = 4

So, |A| = (2)(4) − (3)(1) = 8 − 3 = 5.

Since |A| ≠ 0, the matrix A is invertible.

Subject: Mathematics  |  Chapter: Matrices  |  Difficulty Level: Medium

FAQs: NCERT Solutions for Class 12 Maths Chapter 3 Matrices

1. Is Chapter 3 Matrices important for the Class 12 board exam?

Yes, Matrices is a high-weightage chapter in Class 12 Maths. Questions are frequently asked from matrix operations, properties, and transpose, making it a must-prepare chapter.

2. Are these NCERT Solutions enough for board exam preparation?

Absolutely. If you practice all NCERT questions and examples properly, these solutions are more than sufficient to score well in the board exam.

3. Does Chapter 3 Matrices come in JEE exams?

Yes. Basic concepts of matrices are used in JEE Main and JEE Advanced, especially in combination with determinants and systems of linear equations.

4. How should I study Matrices to score full marks?

• Start by understanding types of matrices

• Practice matrix operations thoroughly

• Focus on properties and transpose

• Revise solved examples and NCERT exercise questions regularly

5. Are these solutions based on the latest CBSE syllabus?

Yes, these NCERT Solutions for Class 12 Maths Chapter 3 are updated as per the latest CBSE curriculum and NCERT textbook.

6. Can I use these solutions for quick revision?

Yes. The step-by-step format makes these solutions perfect for last-minute revision before exams.