Matrices – Chapter 3 | AAI ATC Exam Prep

📌 Why Matrices for AAI ATC?

Matrices are a core topic in the AAI ATC written examination (Mathematics & Reasoning section). Questions from this chapter test your understanding of matrix types, operations, transpose, symmetry, and inverse — all areas frequently asked in competitive papers. Mastering this chapter will significantly boost your score.

"The essence of Mathematics lies in its freedom." — Cantor

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3.1 – What is a Matrix? Order & Elements

A matrix is an ordered rectangular array of numbers or functions. These numbers/functions are called the elements or entries of the matrix.

A matrix having m rows and n columns is called a matrix of order m × n. The element in the i^th row and j^th column is denoted as a_ij.

General Form of an m × n Matrix


A = [a_ij]_m×n


[ a₁₁  a₁₂  a₁₃  ···  a₁ₙ ]

[ a₂₁  a₂₂  a₂₃  ···  a₂ₙ ]

[  ·    ·    ·         ·   ]

[ aₘ₁  aₘ₂  aₘ₃  ···  aₘₙ ]

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[ Image Space ]
Add your diagram showing rows & columns labelled on a matrix

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Total number of elements in an m × n matrix = m × n. To find all possible orders for a given number of elements, find all factor pairs of that number.

🎯 Practice MCQs – Matrix Definition & Order

1If a matrix has 18 elements, which of the following is NOT a possible order?

A) 3 × 6

B) 2 × 9

C) 4 × 5

D) 18 × 1

4 × 5 = 20 ≠ 18. So 4 × 5 is NOT possible. Valid orders: 1×18, 18×1, 2×9, 9×2, 3×6, 6×3. ✅ Answer: C

2In the matrix A = [a_ij]_3×4, the element a₂₃ is in:

A) 3rd row, 2nd column

B) 2nd row, 3rd column

C) 2nd row, 2nd column

D) 3rd row, 3rd column

a_ij means i = row, j = column. So a₂₃ → 2nd row, 3rd column. ✅ Answer: B

3A matrix A = [a_ij]_m×n is a square matrix if:

A) m < n

B) m > n

C) m = n

D) m + n = 0

A square matrix has equal number of rows and columns, i.e., m = n. ✅ Answer: C

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3.2 – Types of Matrices

Column Matrix

Only one column. Order: m × 1

Row Matrix

Only one row. Order: 1 × n

Square Matrix

No. of rows = No. of columns. m = n

Diagonal Matrix

Non-diagonal elements = 0 (b_ij = 0, i ≠ j)

Scalar Matrix

Diagonal matrix where all diagonal elements are equal

Identity Matrix (I)

Diagonal = 1, rest = 0. a_ij = 1 if i=j, else 0

Zero Matrix (O)

All elements are zero. Also called null matrix.

Equal Matrices

Same order AND each a_ij = b_ij

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[ Image Space ]
Add your visual showing examples of each matrix type with 2×2 or 3×3 examples

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Every Identity Matrix is a Scalar Matrix, but every Scalar Matrix is NOT an Identity Matrix. When k = 1 in a scalar matrix, it becomes an identity matrix.

🎯 Practice MCQs – Types of Matrices

1The number of all possible matrices of order 3×3 with each entry 0 or 1 is:

A) 27

B) 18

C) 81

D) 512

A 3×3 matrix has 9 elements. Each can be 0 or 1 → 2 choices each. Total = 2⁹ = 512. ✅ Answer: D

2A diagonal matrix in which all diagonal elements equal √3 is a:

A) Identity matrix

B) Scalar matrix

C) Zero matrix

D) Column matrix

A scalar matrix has b_ij = k (constant) when i = j and 0 otherwise. Here k = √3. Since k ≠ 1, it's not identity. ✅ Answer: B

3If A = [a_ij]_2×2 where a_ij = (i + 2j)²/2, then a₁₂ = ?

A) 4.5

B) 12.5

C) 8

D) 6.5

a₁₂ → i=1, j=2: (1 + 2×2)²/2 = (1+4)²/2 = 25/2 = 12.5 ✅ Answer: B

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3.3 – Operations on Matrices

Addition of Matrices

Two matrices A and B can be added only if they are of the same order. The sum C = A + B is obtained by adding corresponding elements: c_ij = a_ij + b_ij.

Scalar Multiplication

If A = [a_ij]_m×n and k is a scalar, then kA = [k·a_ij]_m×n — every element is multiplied by k.

Matrix Multiplication (AB)

AB is defined only when number of columns of A = number of rows of B. If A is m×n and B is n×p, then AB is m×p.

c_ik = Σ a_ij · b_jk (j = 1 to n)

The (i,k)th element of AB = dot product of ith row of A with kth column of B

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[ Image Space ]
Add your worked example showing 2×2 matrix multiplication step by step

Properties of Matrix Operations

Property	Addition	Multiplication
Commutative	✅ A + B = B + A	❌ AB ≠ BA (generally)
Associative	✅ (A+B)+C = A+(B+C)	✅ (AB)C = A(BC)
Distributive	✅ A(B+C) = AB + AC
Identity	O (Zero Matrix)	I (Identity Matrix)

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Important for AAI ATC: AB = O does NOT imply A = O or B = O. Two non-zero matrices can multiply to give a zero matrix!

🎯 Practice MCQs – Matrix Operations

1If A is a 2×3 matrix and B is a 3×4 matrix, then the order of AB is:

A) 3 × 3

B) 2 × 4

C) 4 × 2

D) 3 × 4

A (2×3) × B (3×4) → inner dimensions (3=3) match, result is m×p = 2×4. ✅ Answer: B

2If A = [[1,0],[0,−1]] and B = [[0,1],[1,0]], what is AB?

A) [[0,1],[−1,0]]

B) [[0,−1],[1,0]]

C) [[1,0],[0,1]]

D) [[0,1],[1,0]]

AB: Row1×Col1 = 0, Row1×Col2 = 1, Row2×Col1 = -1, Row2×Col2 = 0 → [[0,1],[-1,0]] ✅ Answer: A

3If 2A + 3X = 5B, then X = ?

A) (5B + 2A)/3

B) (5B − 2A)/3

C) (5B − 2A)/5

D) 3(5B − 2A)

3X = 5B − 2A → X = (1/3)(5B − 2A). ✅ Answer: B

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3.4 – Transpose of a Matrix

The transpose of matrix A = [a_ij]_m×n is A′ (or Aᵀ) = [a_ji]_n×m — rows and columns are interchanged.

If A is m×n, then A′ is n×m. For example:


If A = [3  5]      then  A′ = [3  √3  0]
       [√3 1]                  [5   1  -1/5]
       [0  -1/5]
(3×2 matrix)              (2×3 matrix)

Properties of Transpose

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(A′)′ = A — Double transpose gives back the original matrix

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(kA)′ = kA′ — Scalar comes out of transpose

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(A + B)′ = A′ + B′ — Transpose distributes over addition

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(AB)′ = B′A′ — Reversal law: transpose of a product = product of transposes in REVERSE order

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[ Image Space ]
Add diagram showing a 3×2 matrix transposed to 2×3 with elements labelled

Symmetric & Skew-Symmetric Matrices

Symmetric Matrix

A′ = A (transpose equals original). a_ij = a_ji for all i, j.

Skew-Symmetric Matrix

A′ = −A. All diagonal elements are zero.

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Key Theorem: Any square matrix A can be expressed as: A = ½(A + A′) + ½(A − A′), where ½(A + A′) is symmetric and ½(A − A′) is skew-symmetric.

🎯 Practice MCQs – Transpose & Symmetry

1If A and B are symmetric matrices of the same order, then (AB − BA) is:

A) Skew-symmetric matrix

B) Symmetric matrix

C) Zero matrix

D) Identity matrix

(AB−BA)′ = (AB)′−(BA)′ = B′A′−A′B′ = BA−AB = −(AB−BA). So it's skew-symmetric. ✅ Answer: A

2For a skew-symmetric matrix A of order 3, what must be true about its diagonal elements?

A) All equal to 1

B) All equal to −1

C) All equal to 0

D) All different

Since A′ = −A, for diagonal: a_ii = −a_ii → 2a_ii = 0 → a_ii = 0 for all i. ✅ Answer: C

3If A = [[cos α, −sin α],[sin α, cos α]] and A + A′ = I, then α = ?

A) π/6

B) π/3

C) π

D) 3π/2

A + A′ gives diagonal = 2cos α. For this to equal I, 2cos α = 1 → cos α = 1/2 → α = π/3. ✅ Answer: B

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3.5 – Invertible Matrices

A square matrix A of order m is invertible if there exists a square matrix B of the same order such that AB = BA = I. B is called the inverse of A, denoted A⁻¹.

The inverse of a square matrix, if it exists, is unique.

If AB = BA = I, then B = A⁻¹ and A = B⁻¹

Also: (AB)⁻¹ = B⁻¹A⁻¹ (Reversal law for inverse)

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[ Image Space ]
Add your example of finding inverse using matrix equation AB = I for a 2×2 matrix

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A rectangular matrix does NOT possess an inverse. Only square matrices can be invertible.

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If B is the inverse of A, then A is also the inverse of B — the relationship is mutual.

🎯 Practice MCQs – Invertible Matrices

1Matrices A and B are inverses of each other if and only if:

A) AB = BA

B) AB = BA = 0

C) AB = 0, BA = I

D) AB = BA = I

By definition of invertible matrices: A·B = B·A = I (Identity). ✅ Answer: D

2If A = [[2,3],[1,2]] is invertible, what is the value of |AB − BA| if B = A⁻¹?

A) 0

B) 1

C) −1

D) 4

If B = A⁻¹, then AB = BA = I. So AB − BA = I − I = O (zero matrix), |O| = 0. ✅ Answer: A

3If A is a square matrix such that A² = A, then (I + A)³ − 7A equals:

A) A

B) I − A

C) I

D) 3A

(I+A)³ = I + 3A + 3A² + A³. Since A²=A, A³=A. So = I + 3A + 3A + A = I + 7A. Thus (I+A)³ − 7A = I. ✅ Answer: C

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Quick Revision – Chapter 3 Summary

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A matrix of order m×n has mn elements. Square matrix: m = n.

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Types: Column, Row, Square, Diagonal, Scalar, Identity, Zero matrices.

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Addition requires same order. Multiplication: columns of A = rows of B.

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Matrix multiplication is NOT commutative (AB ≠ BA in general).

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Transpose: (A′)′=A, (AB)′=B′A′. Symmetric: A′=A. Skew-symmetric: A′=−A.

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Invertible: AB = BA = I. Inverse is unique. (AB)⁻¹ = B⁻¹A⁻¹.

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Any square matrix = symmetric part + skew-symmetric part.

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Chapter 3 – Matrices