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Mathematics · Class 12 Science

Ch 3Matrices — Class 12 Mathematics, concept-first.

In many real situations we need to record several numbers that are all related to each other in a grid-like way -- for instance, the marks scored by students in subjects, or the number of items of kinds sent from warehouses to shops.

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Key concepts

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Symmetric And Skew Symmetric Matrices

These are two special kinds of square matrices, defined by how a matrix compares with its own transpose (the matrix with rows and columns swapped). They are among the most-tested ideas in the Matrices chapter.

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1 concept · 3 appearances

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

1

Concept and Notation of a Matrix

In many real situations we need to record several numbers that are all related to each other in a grid-like way -- for instance, the marks scored by students in subjects, or the number of items of kin…

2

Order and Types of Matrices

If a matrix has rows and columns, we say is a matrix of order (read " by "), and it has exactly elements in all -- the row-count is always stated first.

3

Equality of Matrices

Two matrices cannot be compared entry by entry unless they are laid out the same way, so equality of matrices needs two conditions to both hold at once, not just one.

4

Addition and Scalar Multiplication of Matrices

Definition. If and are two matrices of the same order , their sum is the matrix obtained by adding corresponding entries: Addition is defined only when the two matrices have identical order; is simply…

5

Multiplication of Matrices

Unlike addition, matrix multiplication does not require the two matrices to have the same order -- but it does require a different kind of match.

6

Transpose of a Matrix

Definition. The transpose of a matrix of order is the matrix obtained by interchanging its rows and columns: the th row of becomes the th column of the transpose, and vice versa.

7

Symmetric and Skew-Symmetric Matrices

Both symmetric and skew-symmetric matrices are defined only for square matrices, since the definitions compare a matrix to its own transpose, and and can only be equal (or negatives of each other) whe…

8

Non-Commutativity and Zero-Divisors in Matrix Multiplication

Unlike addition of matrices and unlike ordinary multiplication of real numbers, matrix multiplication does not, in general, satisfy -- even when both products are defined and have the same order.

9

Invertible Matrices and Uniqueness of Inverse

Definition. A square matrix of order is said to be invertible (or non-singular) if there exists a square matrix of the same order such that where is the identity matrix of order (Section 2).

Summary

This chapter developed the algebra of matrices with real entries from first principles. A matrix was defined as a rectangular array of order , and the standard shape- and pattern-based types were clas…

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 20 questions20 questions
  1. Q1A = [[8, 0], [4, -2]] and B = [[2, -2], [-5, 1]], find another matrix X where 2A + 3X = 5B.Preview
  2. Q2If A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]], then show that A^2 - 4A - 5 I3 = 0. **OR** If A = [[i, -i], [-i, i]] and B = [[1, -1], [-1, 1]], t…Preview
  3. Q3If A = (1 5; 6 7) [2×2 matrix, row1: 1,5; row2: 6,7], then show that A - Aᵀ is a skew symmetric matrix.Preview
  4. Q4If (2 1; 3 4) × (x; y) = (1; -1) [i.e. 2x + y = 1 and 3x + 4y = -1], then find the values of x and y.Preview
  5. Q5If A = (x -2; 2 1), B = (3 4; 0 1), C = (-1 -2; y 2) [all 2×2 matrices] and A + B = BC, then find the values of x and y. **OR** If A + 2B =…Preview
  6. Q6Show that A = [[1, 2, 2], [2, 1, 2], [2, 2, 1]] matrix satisfies the equation A² - 4A - 5I₃ = 0. Hence find A⁻¹. [I₃ = [[1,0,0],[0,1,0],[0,0…Preview
  7. Q7A = (a; 6) and B = (3; b) [2×1 column matrices] and A = B, then (a, b) is equal to: **OR** If a + b + c = 0, then value of |a b c; b c a; c…Preview
  8. Q8If [5 3; x 7] = [y 3; 7 x], then (x+y) is equal to (a) 5 (b) 12 (c) 7 (d) 14Preview
  9. Q9If A = (1 3; 2 -2), find AA^T.Preview
  10. Q10If 2X + 3Y = (2 3; 4 0) and 3X + 2Y = (2 -2; -1 5), find X and Y. **OR** If A = (3 1; -1 2), show that A² - 5A + 7I = 0.Preview
  11. Q11If 2A^T + B = (2 5; 10 2) and 2B^T + A = (1 8; 4 1), find the value of matrix A.Preview
  12. Q12If A + I₃ = (1 3 4; -1 1 3; -2 -3 1), find (A + I₃)(A - I₃). **OR** If 3A = (-1 2 -2; -2 1 2; 2 2 1), then show that A is an orthogonal matr…Preview
  13. Q13If the matrix (4 -k; -2 3) has no inverse matrix, then the value of k is (a) 6 (b) -6 (c) 12 (d) -12Preview
  14. Q14If F(x) = (cosx -sinx 0; sinx cosx 0; 0 0 1), then show that F(x).F(y) = F(x+y). **OR** If A = (3 1; 0 2), then show that (A⁻¹)ᵀ = (Aᵀ)⁻¹ wh…Preview
  15. Q15If $A = [a_{ij}]$ is a $2 \times 2$ matrix, where $a_{ij} = \dfrac{1}{2}(i + 2j)^2$, then $A$ is (a) $\begin{bmatrix} \dfrac{9}{2} & \dfrac{…Preview
  16. Q16If $A = \begin{bmatrix} 1 & 2 \\ 2 & 1 \end{bmatrix}$ and $f(x) = x^2 - 2x - 5$, then $f(A)$ is equal to (a) $\begin{bmatrix} -2 & 0 \\ 0 &…Preview
  17. Q17Let $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 1 \\ -2 & 0 \end{bmatrix}$. Match the matrix on the lef…Preview
  18. Q18If $X = \begin{bmatrix} 3 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 3 \end{bmatrix}$, then $X^5$ will be (a) $36X$ (b) $50X$ (c) $90X$ (d) $81X$Preview
  19. Q19Let $S$ be a $2 \times m$ ordered and $T$ be a $3 \times n$ ordered matrix, and conformable for product $TS$ matrix of order $p \times 4$. T…Preview
  20. Q20The values of $a$, $b$ and $c$ for which the matrix $\begin{bmatrix} 1 & a+b-c & a+b+c \\ 1 & 2 & a-b+c \\ 9 & 5 & 3 \end{bmatrix}$ will be…Preview

More questions

32 Q
+Show 3 questions3 questions
  1. Q30If $A$ is any square matrix, prove that $A+A^T$ is symmetric and $A-A^T$ is skew-symmetric.Free
  2. Q31If $A$ and $B$ are symmetric matrices of the same order such that $AB=BA$, prove that $AB$ is symmetric.Preview
  3. Q32For $A=\begin{bmatrix}1&2\\2&1\end{bmatrix}$, verify that $A$ is symmetric, and find $A^{-1}$.Preview
+Show 9 questions9 questions
  1. Example 1A matrix $A=[a_{ij}]$ of order $2\times 3$ is defined by $a_{ij}=i+j$. Write down the matrix $A$ and state its order and the number of eleme…Free
  2. Example 2Classify each of the following matrices as a row matrix, column matrix, square matrix, diagonal matrix, scalar matrix, identity matrix or ze…Free
  3. Example 3Find $x$ and $y$ if $\begin{bmatrix}x+y & 2\\5 & x-y\end{bmatrix}=\begin{bmatrix}6&2\\5&2\end{bmatrix}$.Free
  4. Example 4If $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}5&6\\7&8\end{bmatrix}$, find $3A-2B$.Preview
  5. Example 5If $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ and $B=\begin{bmatrix}2&0\\1&3\end{bmatrix}$, find $AB$ and $BA$, and show that $AB\neq BA$.Preview
  6. Example 6Show that $A=\begin{bmatrix}1&0\\0&0\end{bmatrix}$ and $B=\begin{bmatrix}0&0\\0&1\end{bmatrix}$ are both non-zero matrices, yet $AB$ is the…Preview
  7. Example 7If $A=\begin{bmatrix}1&2&3\\4&5&6\end{bmatrix}$ and $B=\begin{bmatrix}0&1&-1\\2&3&5\end{bmatrix}$, find $A^T$ and $B^T$, and verify that $(A…Preview
  8. Example 8Express $A=\begin{bmatrix}2&3\\5&7\end{bmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.Preview
  9. Example 9Find the inverse of $A=\begin{bmatrix}2&1\\1&1\end{bmatrix}$ using the adjoint-determinant method, and verify that $AA^{-1}=I$.Preview
+Show 4 questions4 questions
  1. Q10A matrix has $24$ elements. Find all the possible orders it can have, and state how many of these orders give a square matrix.Free
  2. Q11Construct the $2\times 3$ matrix $A=[a_{ij}]$ whose elements are given by $a_{ij}=(i-j)^2$.Free
  3. Q12Classify each of the following matrices as a row, column, diagonal, scalar, identity, or zero matrix: (i) $\begin{bmatrix}7\end{bmatrix}$ (i…Preview
  4. Q13State, with a brief reason, whether the following statement is true or false: \textit{\"Every scalar matrix is a diagonal matrix, but a diag…Preview
+Show 4 questions4 questions
  1. Q14If $A=\begin{bmatrix}2&-1\\0&3\end{bmatrix}$ and $B=\begin{bmatrix}1&4\\-2&5\end{bmatrix}$, find $A+B$.Free
  2. Q15For the matrices $A$ and $B$ of Q1, find $2A-3B$.Free
  3. Q16For the matrices $A$ and $B$ of Q1, find the matrix $X$ such that $A+X=B$.Preview
  4. Q17For the matrices $A$ and $B$ of Q1, verify that $A+B=B+A$.Preview
+Show 4 questions4 questions
  1. Q18If $A=\begin{bmatrix}3&1\\2&4\end{bmatrix}$ and $B=\begin{bmatrix}1&2\\0&5\end{bmatrix}$, find $AB$.Free
  2. Q19For the matrices $A$ and $B$ of Q1, find $BA$ and check whether $AB=BA$.Free
  3. Q20If $A=\begin{bmatrix}1&2&0\\3&-1&4\end{bmatrix}$ and $B=\begin{bmatrix}2&1\\0&3\\-1&5\end{bmatrix}$, find $AB$ and state its order.Preview
  4. Q21Show that $A=\begin{bmatrix}2&4\\1&2\end{bmatrix}$ and $B=\begin{bmatrix}2&-4\\-1&2\end{bmatrix}$ are both non-zero matrices, yet $AB$ is th…Preview
+Show 4 questions4 questions
  1. Q22Find the transpose of $A=\begin{bmatrix}3&-1&2\\0&4&5\end{bmatrix}$.Free
  2. Q23Determine whether $M=\begin{bmatrix}0&3&-2\\-3&0&5\\2&-5&0\end{bmatrix}$ is symmetric, skew-symmetric, or neither. Justify your answer.Free
  3. Q24Express $A=\begin{bmatrix}4&6\\2&8\end{bmatrix}$ as the sum of a symmetric matrix and a skew-symmetric matrix.Preview
  4. Q25Prove that every diagonal entry of a skew-symmetric matrix must be zero.Preview
+Show 4 questions4 questions
  1. Q26Find the inverse of $A=\begin{bmatrix}3&2\\1&1\end{bmatrix}$ using the adjoint-determinant method.Free
  2. Q27For the matrix $A$ and its inverse found in Q1, verify that $AA^{-1}=A^{-1}A=I$.Free
  3. Q28Show that the matrix $A=\begin{bmatrix}2&4\\1&2\end{bmatrix}$ is singular, and explain why it has no inverse.Preview
  4. Q29Two students claim that the matrix $A=\begin{bmatrix}1&2\\3&4\end{bmatrix}$ has two different inverses, $B=\begin{bmatrix}-2&1\\1.5&-0.5\end…Preview