Class 12 JEE Matrices Important Questions and Solutions by Physics Wallah

Matrices questions often become difficult when a familiar formula is applied without checking its conditions. The order of two matrices determines whether their product exists, a zero determinant rules out an inverse, and the order of multiplication matters when matrices do not commute. Questions involving adjoints, characteristic equations and high powers also require you to recognise which property can shorten the calculation.

Important questions by PW move from matrix order and special types through multiplication, transpose, determinants, inverse, orthogonality, matrix powers and systems of linear equations. With solved examples, PW helps you identify the relevant property, apply the correct identity and calculate only what the question asks for, reducing unnecessary steps and avoiding common errors in JEE Physics. 

Matrices: Important Questions and Answers

Q1. What Is a Matrix, and How Do You Find Its Order and Number of Elements?

A matrix is a rectangular arrangement of elements in rows and columns. Its order is written as m × n, where m is the number of rows and n is the number of columns.

The total number of elements is:

Number of elements = sum n

For an element aᵢⱼ, i represents the row number, and j represents the column number.

A matrix differs from a determinant in the following ways:

• A matrix may be rectangular or square.

• A determinant is defined only for a square matrix.

• A matrix is an arrangement of elements, whereas a determinant evaluates to a number.

Solved example: A matrix has 2 rows and 3 columns. Find its order and number of elements.

Order = 2 × 3

Number of elements = 2 × 3

Answer: Order = 2 × 3; number of elements = 6.

JEE tip: Always state the order as m × n, not just the product mn. Two matrices can contain the same number of elements but have different orders.

Q2. How Do You Find the Number of Possible Orders of a Matrix?

If a matrix contains N elements, its possible orders are the ordered pairs of positive integers (m, n) satisfying:

mn = N

If the prime factorisation of N is

N = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ,

the number of possible orders is:

(a₁ + 1)(a₂ + 1)…(aₖ + 1)

This follows because each positive divisor of N can serve as the number of rows, with the corresponding quotient giving the number of columns.

Solved example: Find the number of possible orders of a matrix containing 4,800 elements.

Prime factorisation:

4800 = 2⁶ × 3¹ × 5²

Therefore,

Number of orders = (6 + 1)(1 + 1)(2 + 1)

= 7 × 2 × 3

= 42

Memory tip: For N elements, count the positive divisors of N. Treat (m, n) and (n, m) as different orders when m ≠ n.

Q3. What Are the Different Types of Matrices and Their Properties?

Matrices can be classified according to their shape and the properties of their elements.

Shape-based matrices

• Row matrix: Has order 1 × n.

• Column matrix: Has order m × 1.

• Null matrix: Every element is zero.

• Horizontal matrix: Has fewer rows than columns, m < n.

• Vertical matrix: Has more rows than columns, m > n.

• Square matrix: Has an equal number of rows and columns.

Special square matrices

• Diagonal matrix: Every off-diagonal element is zero, so aᵢⱼ = 0 for i ≠ j.

• Scalar matrix: A diagonal matrix whose diagonal elements are equal.

• Identity matrix: A diagonal matrix whose diagonal elements are all 1.

• Upper triangular matrix: aᵢⱼ = 0 whenever i > j.

• Lower triangular matrix: aᵢⱼ = 0 whenever i < j.

Their inclusion relationship is:

Identity ⊂ Scalar ⊂ Diagonal ⊂ Triangular ⊂ Square

For an n × n matrix:

• A triangular matrix has n(n − 1)/2 forced zeros.

• A diagonal matrix has n(n − 1) forced zeros.

• A non-null matrix can have at most n² − 1 zero elements.

Solved example: How many forced zeros does a 4 × 4 upper triangular matrix have?

n(n − 1)/2 = 4(4 − 1)/2

= 6

These are the entries below the principal diagonal.

JEE tip: Identify the matrix type before calculating its determinant or inverse. Special matrices often have properties that make lengthy calculations unnecessary.

Q4. How Do You Calculate the Determinant and Trace of a Matrix?

For a diagonal, upper triangular or lower triangular matrix, the determinant equals the product of the principal-diagonal elements:

det(A) = a₁₁ × a₂₂ × … × aₙₙ

A square matrix is classified as:

• Singular: det(A) = 0

• Non-singular: det(A) ≠ 0

The trace of a square matrix is the sum of its principal-diagonal elements:

tr(A) = a₁₁ + a₂₂ + … + aₙₙ

Important trace properties include:

tr(A + B) = tr(A) + tr(B)

tr(λA) = λ tr(A)

For square matrices A and B of the same order:

tr(AB) = tr(BA)

Solved example: Find the determinant and trace of

A = [ 2  3  1 ]

    [ 0  4  5 ]

    [ 0  0  6 ]

Since A is upper triangular:

det(A) = 2 × 4 × 6 = 48

The trace is:

tr(A) = 2 + 4 + 6 = 12

JEE tip: For a triangular matrix, use the diagonal entries directly to find the determinant. For trace questions, ignore all off-diagonal elements.

Q5. When Can You Add Matrices, and How Does Scalar Multiplication Work?

Two matrices can be added or subtracted only when they have the same order. Their corresponding elements are added or subtracted.

If A = [aᵢⱼ] and B = [bᵢⱼ], then:

(A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ

Matrix addition is commutative and associative:

A + B = B + A

(A + B) + C = A + (B + C)

Scalar multiplication means multiplying every element of a matrix by the same scalar:

λA = [λaᵢⱼ]

For an n × n matrix:

det(λA) = λⁿ det(A)

Solved example: If A is a 3 × 3 matrix with det(A) = 5, find det(2A).

det(2A) = 2³ × det(A)

= 8 × 5

= 40

Memory tip: When a scalar multiplies an n × n matrix, its effect on the determinant is raised to the power n.

Q6. When Is Matrix Multiplication Possible, and What Properties Does It Follow?

Suppose A has order m × n and B has order n × p.

The product AB exists because the number of columns of A equals the number of rows of B. Its order is m × p.

(m × n)(n × p) = m × p

Each element of AB is obtained by multiplying a row of A by a column of B and adding the products.

Matrix multiplication is generally not commutative:

AB ≠ BA

However, it is associative and distributive:

(AB)C = A(BC)

A(B + C) = AB + AC

For square matrices of the same order:

det(AB) = det(A) × det(B)

The identity matrix satisfies:

AI = IA = A, and Iᵏ = I

for every positive integer k.

Also, AB = O does not necessarily imply A = O or B = O. Two non-null matrices can have a null product.

Solved example: Let

A = [ 1  2 ]      B = [ 2  0 ]

    [ 3  4 ]          [ 1  3 ]

Find AB.

AB = [ 1(2) + 2(1)   1(0) + 2(3) ]

     [ 3(2) + 4(1)   3(0) + 4(3) ]

Answer:

AB = [ 4   6  ]

     [ 10  12 ]

JEE tip: Check the inner dimensions first. If both AB and BA exist, do not assume their values are equal.

Q7. What Are the Properties of the Transpose of a Matrix?

The transpose of a matrix is obtained by interchanging its rows and columns. If A has order m × n, then Aᵀ has order n × m.

Important properties are:

(A + B)ᵀ = Aᵀ + Bᵀ

(AB)ᵀ = BᵀAᵀ

(Aᵀ)ᵀ = A

det(Aᵀ) = det(A)

The order of multiplication reverses when taking the transpose of a product.

Solved example: If A is a 2 × 3 matrix, what is the order of Aᵀ?

Interchanging rows and columns gives:

Order of Aᵀ = 3 × 2

JEE tip: Remember that (AB)ᵀ = BᵀAᵀ, not AᵀBᵀ. This reversal is frequently tested in matrix-identity questions.

Q8. What Are Symmetric and Skew-Symmetric Matrices?

A square matrix A is symmetric if:

A = Aᵀ

A square matrix A is skew-symmetric if:

A = −Aᵀ

For a symmetric matrix, corresponding elements on opposite sides of the principal diagonal are equal. For a skew-symmetric matrix, they are negatives of each other.

Every diagonal element of a skew-symmetric matrix is zero. Furthermore, the determinant of a skew-symmetric matrix of odd order is zero.

For any square matrix A:

A + Aᵀ is symmetric

A − Aᵀ is skew-symmetric

Every square matrix can be decomposed into symmetric and skew-symmetric parts:

A = ½(A + Aᵀ) + ½(A − Aᵀ)

If A and B are symmetric matrices of the same order, then:

• AB + BA is symmetric.

• AB − BA is skew-symmetric.

Solved example: If

A = [ 2  5 ]

    [ 1  3 ]

find the symmetric and skew-symmetric parts of A.

First,

Aᵀ = [ 2  1 ]

     [ 5  3 ]

The symmetric part is:

½(A + Aᵀ) = ½ [ 4  6 ]  =  [ 2  3 ]

              [ 6  6 ]     [ 3  3 ]

The skew-symmetric part is:

½(A − Aᵀ) = ½ [  0  4 ]  =  [  0  2 ]

              [ −4  0 ]     [ −2  0 ]

Adding these two parts gives the original matrix A.

Memory tip: To identify the two parts quickly, add A and Aᵀ for the symmetric part and subtract Aᵀ from A for the skew-symmetric part. Divide both results by 2.

Q9. What Is an Orthogonal Matrix, and What Are Its Properties?

A square matrix A is orthogonal if:

AAᵀ = AᵀA = I

Therefore, its inverse is its transpose:

A⁻¹ = Aᵀ

Taking determinants of AAᵀ = I:

det(A) × det(Aᵀ) = 1

Since det(Aᵀ) = det(A):

[det(A)]² = 1

Hence,

det(A) = ±1

The rows and columns of an orthogonal matrix have unit length and are mutually orthogonal in pairs.

Solved example: Verify that

A = [ 0  −1 ]

    [ 1   0 ]

is orthogonal and find its determinant.

Its transpose is:

Aᵀ = [  0  1 ]

     [ −1  0 ]

Multiplying:

AAᵀ = [ 1  0 ] = I

      [ 0  1 ]

Thus, A is orthogonal.

Its determinant is:

det(A) = 0(0) − (−1)(1) = 1

JEE tip: If a question establishes that a matrix is orthogonal, use A⁻¹ = Aᵀ instead of calculating the inverse using cofactors.

Q10. What Is the Adjoint of a Matrix, and How Is It Related to the Inverse?

The adjoint of a square matrix is the transpose of its cofactor matrix:

adj(A) = [Cᵢⱼ]ᵀ

The fundamental identity is:

A · adj(A) = adj(A) · A = det(A) · I

For an n × n matrix, the following identities are useful:

det(adj(A)) = [det(A)]ⁿ⁻¹

For n ≥ 2:

adj(adj(A)) = [det(A)]ⁿ⁻² · A

For square matrices A and B of the same order:

adj(AB) = adj(B) · adj(A)

For a scalar λ:

adj(λA) = λⁿ⁻¹ · adj(A)

Solved example: If A is a 3 × 3 matrix and det(A) = 4, find det(adj(A)).

Using the determinant identity:

det(adj(A)) = [det(A)]³⁻¹

= 4²

= 16

JEE tip: When a question gives the determinant of a matrix and asks for the determinant of its adjoint, use the identity directly instead of calculating cofactors.

Q11. When Does a Matrix Have an Inverse, and Which Inverse Identities Should You Remember?

A square matrix A is invertible if and only if it is non-singular:

det(A) ≠ 0

Its inverse is:

A⁻¹ = adj(A) / det(A)

Other useful identities include:

(Aᵀ)⁻¹ = (A⁻¹)ᵀ

(A⁻¹)⁻¹ = A

(Aᵏ)⁻¹ = (A⁻¹)ᵏ

det(A⁻¹) = 1 / det(A)

The inverse of a product, when the matrices are invertible, follows the reverse order:

(AB)⁻¹ = B⁻¹A⁻¹

Solved example: If A is a non-singular matrix with det(A) = 5, find det(A⁻¹).

det(A⁻¹) = 1 / det(A)

= 1/5

Memory tip: Always check that the determinant is non-zero before applying an inverse formula. For products, remember that the order reverses.

Q12. What Is the Cayley–Hamilton Theorem, and How Does It Simplify Matrix Powers?

The characteristic equation of a square matrix A is obtained by setting:

det(A − λI) = 0

The Cayley–Hamilton theorem states that every square matrix satisfies its own characteristic equation.

For a 2 × 2 matrix:

A² − tr(A) · A + det(A) · I = O

This identity expresses A² in terms of A and I, helping reduce higher powers of the matrix.

Solved example: If a 2 × 2 matrix satisfies

A² − 2A + I = O,

find A⁵.

Rearrange the equation:

A² − 2A + I = O

(A − I)² = O

Let N = A − I. Then N² = O and A = I + N.

Using the binomial expansion:

A⁵ = (I + N)⁵

All terms containing N² or higher powers vanish. Therefore,

A⁵ = I + 5N

Substituting N = A − I:

A⁵ = I + 5(A − I)

= 5A − 4I

JEE tip: First find the characteristic equation or the given polynomial identity. Then reduce higher powers using the relation rather than multiplying the matrix repeatedly.

Matrices becomes easier when you identify the property before beginning a calculation. Check dimensions before multiplying, use diagonal entries for triangular-matrix determinants, reverse the order when transposing or inverting products, and apply Cayley–Hamilton or special matrix identities to simplify high powers. Practise the examples on adjoints, orthogonal matrices and linear systems until you can recognise the shortest valid method. PW’s Matrices questions help you revise these concepts and apply them accurately in JEE Mathematics.

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