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Mathematics and Statistics · Class 12 Commerce

Ch 2Matrices — Class 12 Mathematics and Statistics, concept-first.

A matrix is a rectangular arrangement of numbers set out in horizontal rows and vertical columns and enclosed in brackets. Each number in the arrangement is an element (or entry) of the matrix.

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Algebra of Matrices

Matrices of the same order are added or subtracted entry by entry: . A scalar multiplies every entry: .

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Chapter contents

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1

Matrices and Their Types

A matrix is a rectangular arrangement of numbers set out in horizontal rows and vertical columns and enclosed in brackets. Each number in the arrangement is an element (or entry) of the matrix.

2

Algebra of Matrices — Addition, Scalar Multiplication and Multiplication

Addition and subtraction. Two matrices can be added (or subtracted) only when they are of the same order; the sum is obtained by adding corresponding elements. If and are both , then .

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Transpose, Symmetric and Skew-Symmetric Matrices

Transpose. The transpose of a matrix , written (or ), is obtained by interchanging its rows and columns — the first row becomes the first column, the second row the second column, and so on.

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Elementary Transformations of a Matrix

An elementary transformation (or elementary operation) is one of three simple changes applied to the rows or columns of a matrix.

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Inverse of a Matrix — by Adjoint and by Elementary Transformations

For a square matrix , an inverse is a matrix (of the same order) such that where is the unit matrix. A square matrix has an inverse if and only if its determinant is non-zero; such a matrix is called…

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Solving Systems of Linear Equations using Matrices

A system of linear equations can be written as a single matrix equation , where is the matrix of coefficients, the column of unknowns, and the column of constants.

Exercises

+Show 7 questions7 questions
  1. Q12If $A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B=\begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}$, find $AB$.Free
  2. Q13Find the inverse of $A=\begin{pmatrix} 3 & 1 \\ 2 & 1 \end{pmatrix}$ by the adjoint method.Free
  3. Q14Solve the equations $2x+y=5$ and $3x+2y=8$ by the matrix inversion method.Free
  4. Q15If $A$ is a matrix of order $2\times 3$ and $B$ is a matrix of order $3\times 4$, then the order of the product $AB$ is: (a) $2\times 4$ (b)…Preview
  5. Q16A square matrix $A$ is said to be skew-symmetric if: (a) $A^{\mathsf{T}}=A$ (b) $A^{\mathsf{T}}=-A$ (c) $A=A^{-1}$ (d) $|A|=0$Preview
  6. Q17Express $A=\begin{pmatrix} 3 & 5 \\ 1 & 7 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.Preview
  7. Q18Solve the system $x+2y+3z=14$, $\ 2x-y+z=3$, $\ 3x+y-z=2$ by the reduction method.Preview

Sample & Board Papers

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

+Show 16 questions16 questions
  1. Q1If $A = \begin{bmatrix} 2 & 3 \\ a & 6 \end{bmatrix}$ is a singular matrix, then $a =$ ______. (a) 6 (b) $-5$ (c) 3 (d) 4Preview
  2. Q2Find $x, y, z$ if $\left\{5\begin{bmatrix} 0 & 1 \\ 1 & 0 \\ 1 & 1 \end{bmatrix} - \begin{bmatrix} 2 & 1 \\ 3 & -2 \\ 1 & 3 \end{bmatrix}\ri…Preview
  3. Q3Find the inverse of the matrix A by using adjoint method. where $A = \begin{bmatrix} -3 & -1 & 1 \\ 0 & 0 & 1 \\ -15 & 6 & -6 \end{bmatrix}$Preview
  4. Q4Express the following equations in matrix form and solve them by the method of reduction: $x + 2y + z = 8$, $2x + 3y - z = 11$, $3x - y - 2z…Preview
  5. Q5If A = $\begin{bmatrix}4 & 3 & 2\\-1 & 2 & 0\end{bmatrix}$, B = $\begin{bmatrix}1 & 2\\-1 & 0\\1 & -2\end{bmatrix}$ Find $(AB)^{-1}$ by adjo…Preview
  6. Q6If $x + y + z = 3$, $x + 2y + 3z = 4$, $x + 4y + 9z = 6$, then $(y, z) =$ _______ (a) $(-1, 0)$ (b) $(1, 0)$ (c) $(1, -1)$ (d) $(-1, 1)$Preview
  7. Q7If $A = \begin{bmatrix} 7 & 3 & 0 \\ 0 & 4 & -2 \end{bmatrix}$, $B = \begin{bmatrix} 0 & -2 & 3 \\ 2 & 1 & -4 \end{bmatrix}$ then find $A^T…Preview
  8. Q8Find the inverse of $\begin{bmatrix} 3 & 1 & 5 \\ 2 & 7 & 8 \\ 1 & 2 & 5 \end{bmatrix}$ by adjoint method.Preview
  9. Q9State whether the following statement is true or false: If A is a matrix and K is a constant, then $(KA)^T = KA^T$Preview
  10. Q10Find x, y, z, if $\left\{5\begin{bmatrix}0 & 1\\1 & 0\\1 & 1\end{bmatrix} - \begin{bmatrix}2 & 1\\3 & -2\\1 & 3\end{bmatrix}\right\} \begin{…Preview
  11. Q11Solve the following equation by the method of inversion. $2x - y + z = 1$, $x + 2y + 3z = 8$, $3x + y - 4z = 1$Preview
  12. Q12Find the inverse of $\begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$ by adjoint method.Preview
  13. Q13Express the following equations in matrix form and solve them by the method of reduction: $x - y + z = 1$, $2x - y = 1$, $3x + 3y - 4z = 2$…Preview
  14. Q14Solve the following equations by the method of reduction: $x + 3y = 2$, $3x + 5y = 4$Preview
  15. Q15Find the values of $x$ and $y$ from the equation: $[-1,\ 1,\ 4]\left\{ 2\begin{bmatrix} 5 & 5 \\ 6 & 6 \\ -1 & 2 \end{bmatrix} + 3\begin{bma…Preview
  16. Q16Find the adjoint of the matrix $\begin{bmatrix} 1 & -1 & 2 \\ -2 & 3 & 5 \\ -2 & 0 & -1 \end{bmatrix}$.Preview

More questions

+Show 11 questions11 questions
  1. Example 1For the matrix $A=\begin{pmatrix} 4 & -2 & 7 \\ 1 & 0 & 3 \end{pmatrix}$, state (i) its order, (ii) the elements $a_{12}$ and $a_{23}$, and…Free
  2. Example 2If $A=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B=\begin{pmatrix} 0 & -1 \\ 2 & 5 \end{pmatrix}$, find $2A-3B$.Free
  3. Example 3If $A=\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ and $B=\begin{pmatrix} 1 & 0 \\ 2 & 5 \end{pmatrix}$, find $AB$ and $BA$, and hence veri…Free
  4. Example 4Find the transpose of $A=\begin{pmatrix} 2 & -1 & 5 \\ 0 & 3 & 4 \end{pmatrix}$, and state its order.Preview
  5. Example 5Show that $A=\begin{pmatrix} 0 & 2 & -3 \\ -2 & 0 & 4 \\ 3 & -4 & 0 \end{pmatrix}$ is skew-symmetric.Preview
  6. Example 6Express $A=\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.Preview
  7. Example 7Find the inverse of $A=\begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$ by the adjoint method.Preview
  8. Example 8Find the inverse of $A=\begin{pmatrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{pmatrix}$ by the adjoint method.Preview
  9. Example 9Find the inverse of $A=\begin{pmatrix} 2 & 1 \\ 7 & 4 \end{pmatrix}$ using elementary row transformations.Preview
  10. Example 10Solve the equations $x+y=3$ and $2x+3y=8$ by the matrix inversion method.Preview
  11. Example 11Solve the system $x+y+z=6$, $\ 2x+3y-z=5$, $\ 3x-y+z=4$ by the reduction method.Preview