Practice Questions on Matrices

A matrix is a set of numbers arranged in rows and columns to form a rectangular array.

Matrix Practice Problems Covering Core Concepts and Applications

Practice Questions on Matrices - Solved

Question 1: If (A+B)² = A + 2AB + B,² then what can we say about A and B? (Assume AB and BA exist.)
Solution:

(A+B)² = (A+B) (A+B)

⇒ According to question,
⇒ A² + AB + BA + B² = A² + 2AB + B²
⇒ AB+BA = 2AB
⇒ BA = AB

⇒ So, we can say that A and B are commutative

Question 2: If A is an n x m matrix such that AB and BA are both defined, then the order of B is
Solution:

If A size is n x m and it is also given that AB is defined then,

⇒ A_{n x m} X B_{m x ☐} = (AB)_{n x n}
⇒ ☐ = n

OR

⇒ B_{☐ x m} X A_{n x m} = (AB)_{n x n}
⇒ ☐ = n

⇒ So, the size of the matrix B is m x n

Question 3: Under what conditions will the matrix equation A - B² = (A - B) (A + B) be true?
Solution:

We are given, A²-B² = (A-B) (A+B)
⇒ A² - B² = A² + AB - BA + B²
⇒ AB - BA = 0
⇒ AB = BA

⇒ So, we can say that A and B should be commutative

Question 4: If AB = A and BA = B, then show that A and B are idempotent matrices.
Solution:

We are given that,

AB = A
⇒ A(BA) = A
⇒ (AB)A = A
⇒ (A)A = A
⇒ A² = A

⇒ So, we can say that A is idempotent matrix
Similarly, we can prove that B is also an idempotent matrix.

Question 5: Show that the sum of two idempotent matrices A and B is idempotent if AB = BA = 0.
Solution:

We have been given that,

AB = BA = 0 and A² = A and B² = B
⇒ (A+B)² = (A+B) (A+B)
⇒ (A+B)2 = A² + AB + BA + B²
⇒ (A+B)² = A² + B² {since, AB=BA=0}
⇒ (A+B)² = A + B

⇒ Hence, sum of two idempotent matrices A and B is idempotent if AB=BA=0

Question 6: Evaluate \Delta = \begin{vmatrix}1 & \omega & \omega^2\\\omega & \omega^2 & 1\\\omega^2 & 1 & \omega\end{vmatrix} where \omega is one of the cube roots of the unity.
Solution:

Since \omega is the cube root of unity, we know that 1+\omega+\omega^2=0

By applying C₁ → C₁ + C₂ + C₃
⇒\begin{vmatrix}1+\omega+\omega^2 & \omega & \omega^2\\\omega+\omega^2+1 & \omega^2 & 1\\\omega^2+1+\omega & 1 & \omega\end{vmatrix}
Now, we know since 1+\omega+\omega^2=0

Above determinant is written as,
⇒ \begin{vmatrix}0 & \omega & \omega^2\\0 & \omega^2 & 1\\0 & 1 & \omega\end{vmatrix}
⇒ 0

Hence, the value of the given determinant is 0

Question 7: Evaluate \Delta = \begin{vmatrix}a-b & m-n & x-y\\b-c & n-p & y-z\\c-a & p-m & z-x\end{vmatrix}
Solution:

By applying R₁ → R₁ + R₂ + R₃

⇒\Delta = \begin{vmatrix}a-b + (b-c) + (c-a) & m-n + (n-p) + (p-m) & x-y + (y-z) + (z-x)\\b-c & n-p & y-z\\c-a & p-m & z-x\end{vmatrix}
⇒\Delta = \begin{vmatrix}0 & 0 & 0\\b-c & n-p & y-z\\c-a & p-m & z-x\end{vmatrix}
⇒\Delta = 0

Question 8: If A is a symmetric matrix, then prove that adj A (the adjoint of A) is also symmetric.
Solution:

Let 'A' is a symmetric matrix, then A^T = A

We know that,
⇒ (adj A)^T = adj A^T
⇒ (adj A)^T = adj A

Hence, adj A is also a symmetric matrix

Question 9: Show that if A is a non-singular matrix, then det(A⁻¹) = (det(A))⁻¹
Solution:

We know that, |A^-1| = 1 / |A|

⇒ A A^-1 = I_n {where, I is an Identity matrix}
⇒ |A A^-1| = |I_n|
⇒ |A| |A^-1| = 1
⇒ |A^-1| = 1 / |A|
⇒ |A|^-1 = 1 / |A|

Hence proved

Question 10: If A and B are n x n squared matrices and AB = 0 & |B| ≠ 0, then A = 0. State True or False.
Solution:

Since, AB = 0

By multiplying by I on both sides, {where, I is an identity matrix}
⇒ A B B^-1 = 0 B^-1
⇒ A I = 0
⇒ A = 0

Hence the above statement is True.

Practice Questions on Matrices - Unsolved

Question 1: Find x so that \begin{bmatrix}1 & x & 1\\\end{bmatrix} \begin{bmatrix}1 & 3 & 2\\0 & 5 & 1\\0 & 3 & 2\\\end{bmatrix} \begin{bmatrix}1\\1\\x\\\end{bmatrix} = 0

Question 2: Under what conditions is the matrix equation A - B² = (A + B)? Is (A+B) true?

Question 3: Evaluate \Delta = \begin{vmatrix}3 & 2 & 1 & 4\\15 & 29 & 2 & 14\\16 & 19 & 3 & 17\\33 & 39 & 8 & 38\end{vmatrix}

Question 4: Evaluate \Delta = \begin{vmatrix}a & b & c\\b & c & a\\c & a & b\\\end{vmatrix}

Question 5: Prove that \begin{vmatrix}x & a & a & a\\a & x & a & a\\a & a & x & a\\a & a & a & x\\\end{vmatrix} = (x + 3a)(x-a)^3

Question 6: Evaluate \Delta = \begin{vmatrix}1 & bc & a(b+c)\\1 & ca & b(c+a)\\1 & ab & c(a+b)\\\end{vmatrix}

Question 7: If a+b+c = 0, then solve the equation \begin{vmatrix}a-x & c & b\\c & b-x & a\\b & a & c-x\\\end{vmatrix} = 0

Question 8: Prove that if A is idempotent and A ≠ I, then A is singular.

Question 9: Only a square, non-singular matrix possesses an inverse, which is unique. State True or False.

Question 10: If AB = 0, does it imply that it is necessary that BA = 0?