Practice Questions on Matrices

Matrices are rectangular arrays of numbers, symbols, or characters where all of these elements are arranged in each row and column. An array is a collection of items arranged at different locations.

Some of the important formulas and concepts that will help to solve the question based on the matrices are mentioned below.

Order of Matrices

Order of a Matrix tells about the number of rows and columns present in a matrix. The order of a matrix is represented as the number of rows times the number of columns. Let’s say if a matrix has 4 rows and 5 columns then the order of the matrix will be 4⨯5.

Transpose of a Matrix

Transpose of a Matrix is the rearrangement of row elements in column and column elements in a row to yield an equivalent matrix. A matrix in which the elements of the row of the original matrix are arranged in columns or vice versa is called Transpose Matrix. The transpose matrix is represented as AT. if A = [a_ij]_mxn , then A^T = [b_ij]_nxm where b_ij = a_ji.

Properties of the transpose of a matrix are mentioned below:

• (A^T)^T = A
• (A+B)^T = A^T + B^T
• (AB)^T = B^TA^T

Trace of Matrix

Trace of a Matrix is the sum of the principal diagonal elements of a square matrix. Trace of a matrix is only found in the case of a square matrix because diagonal elements exist only in square matrices.

Determinant of a Matrix

Determinant of a matrix is a number associated with that square matrix. The determinant of a matrix can only be calculated for a square matrix. It is represented by |A|. The determinant of a matrix is calculated by adding the product of the elements of a matrix with their cofactors.

Minor of a Matrix

Minor of a matrix for an element is given by the determinant of a matrix obtained after deleting the row and column to which the particular element belongs to. Minor of Matrix is represented by M_ij.

Cofactor of Matrix

Cofactor of a matrix is found by multiplying the minor of the matrix for a given element by (-1)^i+j. Cofactor of a Matrix is represented as Cij. Hence, the relation between the minor and cofactor of a matrix is given as M_ij = (-1)^i+jM_ij.

Must Read

• Minors and Cofactors

Adjoint of Matrix

Adjoint is calculated for a square matrix. Adjoint of a matrix is the transpose of the cofactor of the matrix. The Adjoint of a Matrix is thus expressed as adj(A) = CT where C is the Cofactor Matrix.

Read

• Adjoint of a Matrix

Types of Matrices

Based on the number of rows and columns present and the special characteristics shown, matrices are classified into various types.

• Row Matrix: A Matrix in which there is only one row and no column is called Row Matrix.

• Column Matrix: A Matrix in which there is only one column and now row is called a Column Matrix.

• Horizontal Matrix: A Matrix in which the number of rows is less than the number of columns is called a Horizontal Matrix.

• Vertical Matrix: A Matrix in which the number of columns is less than the number of rows is called a Vertical Matrix.

• Idempotent Matrix: A matrix is said to be idempotent if A² = A

Inverse of a Matrix

A matrix is said to be an inverse of matrix ‘A’ if the matrix is raised to power -1 i.e. A-1. The inverse is only calculated for a square matrix whose determinant is non-zero. The formula for the inverse of a matrix is given as:

A^-1 = adj(A)/det(A) = (1/|A|)(Adj A), where |A| should not be equal to zero, which means matrix A should be non-singular.

Read More

• Inverse of Matrix

Matrices Formulas

Basic formula for the matrices are discussed below:

• A^-1 = adj(A)/|A|

• A(adj A) = (adj A)A = I, where I is an Identity Matrix

• |adj A| = |A|^n-1 where n is the order of matrix A

• adj(adj A) = |A|^n-2A where n is the order of the matrix

• |adj(adj A)| = |A|(n-1)²

• adj(AB) = (adj B)(adj A)

• adj(A^p) = (adj A)^p

• adj(kA) = k^n-1(adj A) where k is any real number

• adj(I) = I

• adj 0 = 0

• If A is symmetric then adj(A) is also symmetric

• If A is a diagonal Matrix then adj(A) is also a diagonal matrix

• If A is a triangular matrix then adj(A) is also a triangular matrix

• If A is a singular Matrix then |adj A| = 0

• (AB)^-1 = B^-1A^-1

Question 1: Find x so that [Tex]\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[/Tex]

Question 2: Under what conditions is the matrix equation A²– B² = (A+B).(A+B) true?

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

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

Question 5: Prove that [Tex]\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[/Tex]

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

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

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

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

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

Question 1: If (A+B)²= A² +2AB+ B² then what can we say about A and B? (Assume AB and BA exists)

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 a nxm matrix such that AB and BA are both defined, then 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 is the matrix equation A²-B² = (A-B) (A+B) will 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 [Tex]\Delta = \begin{vmatrix} 1 & \omega & \omega^2\\ \omega & \omega^2 & 1\\ \omega^2 & 1 & \omega \end{vmatrix}[/Tex] where [Tex]\omega[/Tex] is one of the cube roots of the unity.

Solution:

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

By applying C₁ → C₁ + C₂ + C₃

⇒[Tex]\begin{vmatrix} 1+\omega+\omega^2 & \omega & \omega^2\\ \omega+\omega^2+1 & \omega^2 & 1\\ \omega^2+1+\omega & 1 & \omega \end{vmatrix}[/Tex]

Now, we know since [Tex]1+\omega+\omega^2=0[/Tex]

Above determinant is written as,

⇒ [Tex]\begin{vmatrix} 0 & \omega & \omega^2\\ 0 & \omega^2 & 1\\ 0 & 1 & \omega \end{vmatrix}[/Tex]

⇒ [Tex]0[/Tex]

Hence, the value of the given determinant is 0

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

Solution:

By applying R₁ → R₁ + R₂ + R₃

⇒[Tex]\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}[/Tex]

⇒[Tex]\Delta = \begin{vmatrix} 0 & 0 & 0\\ b-c & n-p & y-z\\ c-a & p-m & z-x \end{vmatrix}[/Tex]

⇒[Tex]\Delta = 0[/Tex]

Question 8: If A is a symmetric matrix, then prove that adj A (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 rowed 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.

What are the necessary conditions for 2 matrix to multiply?

2 matrices could only be multiplied if the number of columns of the first matrix are equal to number of rows of the second matrix.

How to multiply a matrix with a scalar quantity?

If we want to multiply a scalar value “k” , then it will be multiplied with all the elements in the matrix.

When do we say that 2 matrices are equal?

Two matrices A and B are said to be same if they have the same number of rows and columns and each entry of the first matrix is equal to second matrix i.e. A_ij = B_ij

What is the value of the determinant of a matrix if n = m = 1, where n = number of rows and m = number of columns

Value of the determinant of the matrix with n = m = 1, it will the value of the element which is present in the matrix itself.

What is the necessary condition for the inverse of the matrix to exist?

Inverse of the matrix exist only and only if A is non-singular i.e. |A| ≠ 0