Practice Questions

Solve for $x$ given that $\left|\begin{array}{cc} 2 & 3 \ 4 & 5 \end{array}\right| = \left|\begin{array}{cc} x & 3 \ 2x & 5 \end{array}\right|$.

State the condition for a square matrix A to be invertible.

If A is a square matrix of order 3 and $|A| = -4$, then calculate the value of $|2A|$.

In the determinant $\Delta = \left|\begin{array}{ccc} 2 & -3 & 5 \ 6 & 0 & 4 \ 1 & 5 & -7 \end{array}\right|$, calculate the minor of the element $a_{23}$.

Explain what is meant by a 'consistent' and an 'inconsistent' system of linear equations.

List the six ways a determinant of order 3 can be expanded to find its value.

Define a singular matrix.

Create a non-zero $2 \times 2$ singular matrix whose trace (sum of diagonal elements) is 5. Justify your answer.

What is the value of the determinant of a matrix of order 1, $A = [a]$?

Write the formula for the cofactor $A_{ij}$ of an element $a_{ij}$ in terms of its minor $M_{ij}$.

Summarize the conditions used to check the consistency of a system of linear equations $AX = B$. Explain the cases for when the determinant of $A$ is zero and when it is not zero.

Calculate the value of the determinant $\left|\begin{array}{cc} \cos 15^{\circ} & \sin 15^{\circ} \ \sin 75^{\circ} & \cos 75^{\circ} \end{array}\right|$.

For what value of $k$ is the matrix $A = \left[\begin{array}{cc} k & 2 \ 3 & 4 \end{array}\right]$ singular?

Calculate the area of the triangle with vertices at the points $A(-2, -3)$, $B(3, 2)$, and $C(-1, -8)$ using determinants.

If $A = \left[\begin{array}{cc} 3 & -2 \ 4 & -2 \end{array}\right]$, find the value of $k$ such that $A^2 = kA - 2I$, where I is the identity matrix of order 2.

Solve the following system of linear equations using the matrix method: $3x + 2y = 8$ $2x - y = 3$

Calculate the inverse of the matrix $A = \left[\begin{array}{ccc} 2 & -1 & 3 \ 1 & 3 & -1 \ 3 & 2 & 1 \end{array}\right]$.

If $A = \left[\begin{array}{cc} 2 & 3 \ 1 & 2 \end{array}\right]$, demonstrate that $A^2 - 4A + I = O$, where I is the identity matrix and O is the zero matrix of order 2.

If $A$ is a non-singular square matrix of order $n$, what is the formula for $|\text{adj } A|$ in terms of $|A|$?

Explain the difference between $|A|$ for a matrix A and the modulus of a number.

Justify, without expanding, why the determinant of the matrix $A = \begin{vmatrix} \sin^2 x & 1 & \cos^2 x \ \sin^2 y & 1 & \cos^2 y \ \sin^2 z & 1 & \cos^2 z \end{vmatrix}$ is zero.

Evaluate the statement: "If A is a square matrix of order 3 and $|A| = 4$, then $|2 \text{adj}(A)| = 128$". Justify your reasoning.

Derive the condition for three points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ to be collinear using the concept of the area of a triangle expressed as a determinant. Justify your derivation.

Describe the process of finding the adjoint of a $3 \times 3$ matrix $A = [a_{ij}]$.

Justify why the inverse of a diagonal matrix $D = \begin{pmatrix} d_1 & 0 & 0 \ 0 & d_2 & 0 \ 0 & 0 & d_3 \end{pmatrix}$ is $D^{-1} = \begin{pmatrix} 1/d_1 & 0 & 0 \ 0 & 1/d_2 & 0 \ 0 & 0 & 1/d_3 \end{pmatrix}$, provided no diagonal elements are zero.

Let $A$ be a square matrix of order 3. Justify the relationship between $|\text{adj}(A)|$ and $|A|$. Use this to prove that if $A$ is a skew-symmetric matrix of order 3, then $|\text{adj}(A)|=0$.

Formulate a system of linear equations for the following problem and justify the existence of a unique solution: "The sum of three numbers is 20. If we multiply the first number by 2, add the second number to the result, and subtract the third number, we get 23. If we multiply the first number by 3 and add the second and third numbers to it, we get 46." Do not solve the system.

Critique the following argument and identify the flaw: To solve the equation $\begin{vmatrix} x & 2 \ 18 & x \end{vmatrix} = \begin{vmatrix} 6 & 2 \ 18 & 6 \end{vmatrix}$, we can equate the corresponding elements. Thus, $x=6$. Justify your critique with the correct method.

Design a solution using the matrix method to find the cost of three food items. A family of 4 bought 3 kg of rice, 2 kg of sugar, and 5 kg of flour and paid a total of ₹850. Another family of 3 bought 2 kg of rice, 3 kg of sugar, and 4 kg of flour for ₹780. A third family of 5 bought 4 kg of rice, 3 kg of sugar, and 6 kg of flour for ₹1150. Formulate the problem as a system of linear equations and create the complete solution to find the price per kg for each item.

Justify that the value of the determinant $\Delta = \begin{vmatrix} 1 & a & a^2 \ 1 & b & b^2 \ 1 & c & c^2 \end{vmatrix}$ is $(a-b)(b-c)(c-a)$.

State the property that relates the product of a square matrix A and its adjoint with its determinant.

Given the vertices of a triangle $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, write the expression for its area in determinant form. Also, state the condition for the three points to be collinear.

Describe in detail the steps to find the determinant of a $3 \times 3$ matrix $A = \begin{vmatrix} a_{11} & a_{12} & a_{13} \ a_{21} & a_{22} & a_{23} \ a_{31} & a_{32} & a_{33} \end{vmatrix}$ by expanding along the first row ($R_1$). Define the terms 'minor' and 'cofactor' as part of your explanation.

Consider two matrices $P = \begin{pmatrix} 1 & -1 & 0 \ 2 & 3 & 4 \ 0 & 1 & 2 \end{pmatrix}$ and $Q = \begin{pmatrix} 2 & 2 & -4 \ -4 & 2 & -4 \ 2 & -1 & 5 \end{pmatrix}$. First, evaluate the product $PQ$. Then, using this result, design a method to solve the following system of linear equations without finding the inverse of the coefficient matrix directly. $x - y = 3$ $2x + 3y + 4z = 17$ $y + 2z = 7$ Justify your method.

Summarize the steps to solve a system of linear equations in two variables, $a_1x + b_1y = d_1$ and $a_2x + b_2y = d_2$, using the matrix method.

Explain the complete procedure for finding the inverse of a $3 \times 3$ non-singular matrix A. List all the necessary definitions and formulas involved.

Calculate the adjoint of the matrix $A = \left[\begin{array}{ccc} 1 & -2 & 3 \ 0 & 2 & 1 \ -4 & 5 & 2 \end{array}\right]$.

Solve the following system of linear equations using the matrix method: $x + y - z = 3$ $2x + 3y + z = 10$ $3x - y - 7z = 1$

Let $A = \left[\begin{array}{ccc} 1 & -2 & 0 \ 2 & 1 & 3 \ 0 & -2 & 1 \end{array}\right]$. First, calculate $A^{-1}$. Then, use the result to solve the following system of linear equations: $x - 2y = 10$ $2x + y + 3z = 8$ $-2y + z = 7$

The sum of three numbers is 2. If we subtract the second number from twice the first number, we get 3. The sum of the first and third numbers is equal to the second number. Represent this problem as a system of linear equations and solve for the numbers using the matrix method.

Propose a condition on the matrix $B$ for the system of linear equations $AX=B$ to be inconsistent, given that the coefficient matrix $A$ is singular.

If $A$ is a non-singular matrix such that $A^2 - 5A + 7I = O$, design a method to express $A^{-1}$ in terms of $A$ and $I$. Justify each step of your design.

Let $A$ be a square matrix of order $n$. Prove the property $A(\text{adj } A) = (\text{adj } A)A = |A|I$. Using this result, justify the formula for the inverse of a non-singular matrix, $A^{-1} = \frac{1}{|A|}(\text{adj } A)$.

Justify that for any skew-symmetric matrix $A$ of odd order $n$, its determinant must be zero. Critique the statement for a skew-symmetric matrix of even order.