Practice Questions

If $\begin{bmatrix} 2x+y \ 3 \end{bmatrix} = \begin{bmatrix} 7 \ x-y \end{bmatrix}$, calculate the values of $x$ and $y$.

Define a zero matrix (or null matrix). Can a zero matrix be a square matrix? Give an example of a $2 \times 3$ zero matrix.

Construct a $2 \times 2$ matrix $A = [a_{ij}]$ whose elements are given by the formula $a_{ij} = (i - j)^3$.

If matrix A has order $3 \times 4$ and matrix B has order $4 \times 2$, analyze the orders of the product matrices AB and BA to determine which one is defined and what its order is.

Explain the concept of 'order of a matrix'. Given the matrix $A = \begin{bmatrix} \sqrt{3} & 1 & -5 & 17 \ 35 & -2 & \frac{5}{2} & 12 \end{bmatrix}$, identify its order and the total number of elements.

Define a row matrix and give one example.

Justify why for any square matrix A, the matrix A - A' is always a skew-symmetric matrix.

Find the transpose of the matrix $P = \begin{bmatrix} 1 & 0 & -2 \ 4 & 2 & 5 \ 3 & -1 & 7 \end{bmatrix}$.

Create two non-zero 2 \times 2 matrices A and B such that their product AB is a zero matrix.

If a matrix has 7 elements, list all its possible orders.

What is the condition for a matrix $A = [a_{ij}]_{m \times n}$ to be a square matrix?

Summarize the rules for (a) addition of two matrices and (b) multiplication of a matrix by a scalar. Given $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 5 \ 6 & 7 \end{bmatrix}$, find $A+B$ and $3A$.

Express the matrix $A = \begin{bmatrix} 3 & 5 \ 1 & -1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.

If $A = \begin{bmatrix} 2 & 0 \ 1 & 5 \end{bmatrix}$ and $I = \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix}$, find the value of $k$ such that $A^2 = 7A + kI$.

Solve the following matrix equations to find the matrices $X$ and $Y$: $X + Y = \begin{bmatrix} 7 & 0 \ 2 & 5 \end{bmatrix}$ $X - Y = \begin{bmatrix} 3 & 0 \ 0 & 3 \end{bmatrix}$

A school wants to award its students for the values of Honesty, Regularity and Hard work. The number of students to be awarded in each category are 3, 2, and 1 respectively. The award money for each value is ₹1000, ₹1500, and ₹2000 per student. Calculate the total award money required using matrix multiplication.

For the matrices $A = \begin{bmatrix} 1 \ -4 \ 3 \end{bmatrix}$ and $B = \begin{bmatrix} -1 & 2 & 1 \end{bmatrix}$, analyze and verify that $(AB)' = B'A'$.

A trust fund has ₹50,000 to invest in two different types of bonds. The first bond pays 6% interest per year, and the second bond pays 8% interest per year. Using matrix multiplication, formulate a method to determine how to divide the ₹50,000 among the two types of bonds to obtain an annual total interest of ₹3,600. Justify your formulation and solve for the amounts.

State the property for the existence of an additive identity in matrix addition.

Identify the diagonal elements of the matrix $A = \begin{bmatrix} 3 & -1 & 0 \ \frac{3}{2} & 3\sqrt{2} & 1 \ 4 & 3 & -1 \end{bmatrix}$.

Describe the two conditions that must be met for two matrices, $A$ and $B$, to be considered equal. Provide an example of two matrices that have the same order but are not equal.

Define a symmetric matrix and a skew-symmetric matrix. What is the key difference regarding their diagonal elements?

List the four main properties associated with the transpose of matrices.

What is an invertible matrix? State the condition for a matrix $B$ to be the inverse of a square matrix $A$.

Define the following types of matrices, each with a unique example of a $3 \times 3$ matrix: (i) Diagonal Matrix (ii) Scalar Matrix (iii) Identity Matrix Explain the relationship between these three types of matrices.

If $A = \begin{bmatrix} 2 & -1 \ 3 & 1 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \ 0 & -2 \end{bmatrix}$, calculate $3A - 2B$.

Solve for the matrix $X$ if $2X + \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix} = \begin{bmatrix} 3 & 8 \ 7 & 2 \end{bmatrix}$.

Find the real values of $a$ and $b$ if $A = \begin{bmatrix} a & b \ -b & a \end{bmatrix}$ and $A^2 = \begin{bmatrix} 0 & 2 \ -2 & 0 \end{bmatrix}$.

Let $A = \begin{bmatrix} 1 & 2 \ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 4 & 0 \ 1 & -2 \end{bmatrix}$. Calculate $AB$ and $BA$ and demonstrate that matrix multiplication is not commutative.

If A and B are square matrices of the same order, justify why (A+B)^2 = A^2 + 2AB + B^2 is not always true. Propose the necessary condition for this equality to hold.

Evaluate the statement: 'If A is a symmetric matrix, then A^n is also a symmetric matrix for any positive integer n.' Justify your conclusion.

Prove by the principle of mathematical induction that if A = \begin{bmatrix} \cos \theta & \sin \theta \ -\sin \theta & \cos \theta \end{bmatrix}, then A^n = \begin{bmatrix} \cos n\theta & \sin n\theta \ -\sin n\theta & \cos n\theta \end{bmatrix} for all n \in N.

Formulate a 3 \times 3 matrix A which is neither symmetric nor skew-symmetric. Then, create the symmetric matrix P and skew-symmetric matrix Q such that A = P + Q and verify your result.

If A and B are symmetric matrices of the same order, prove that AB - BA is a skew-symmetric matrix.

Propose a proof for the uniqueness of the inverse of a square matrix. That is, if a square matrix A has an inverse, prove that it is unique.

Create a 2 \times 2 matrix A that is its own inverse (A^{-1} = A), but is not the identity matrix I.

Explain the process of multiplying two matrices, $A$ and $B$. What is the essential condition on their orders for the product $AB$ to be defined? If matrix $A$ is of order $m \times n$ and matrix $B$ is of order $n \times p$, what will be the order of the product matrix $AB$?

If A is a square matrix satisfying the equation A^2 - 5A + 7I = O, prove that A is invertible and create an expression for A^{-1} in terms of A and I.

If A is a square matrix such that A^2 = A (an idempotent matrix), prove that (I+A)^3 - 7A = I. Justify each step of the expansion.

If $A = \begin{bmatrix} 1 & 1 & 1 \ 1 & 2 & -3 \ 2 & -1 & 3 \end{bmatrix}$, calculate the value of the expression $A^3 - 6A^2 + 5A + 11I$, where I is the identity matrix of order 3, and analyze the result.

Two families, A and B, have different monthly consumptions of rice, wheat, and sugar (in kg). Family A consumes 10 kg rice, 15 kg wheat, and 4 kg sugar. Family B consumes 12 kg rice, 18 kg wheat, and 5 kg sugar. The prices per kg are ₹40 for rice, ₹30 for wheat, and ₹45 for sugar. (i) Represent the consumption data as a $2 \times 3$ matrix. (ii) Represent the price data as a $3 \times 1$ matrix. (iii) Using matrix multiplication, calculate the monthly expenditure for each family.

Explain why matrix multiplication is not commutative in general. Use a simple example of $2 \times 2$ matrices to illustrate your explanation.

Let A, B, and C be matrices of suitable orders such that their products are defined. Prove the distributive law A(B+C) = AB + AC.

Show that the matrix B'AB is symmetric or skew-symmetric according as A is symmetric or skew-symmetric. Justify your proof for both cases.

If A is an invertible square matrix, justify that (A')^{-1} = (A^{-1})'.