Engineering Mathematics 01 | Linear Algebra Part 1 | Matrix | GATE - For All Branches

Main ideas and lessons (Engineering Mathematics: Linear Algebra—Matrices)

The video introduces a GATE-focused series covering Engineering Mathematics in three parts:

• Linear Algebra
• Calculus
• Probability & Statistics

Within Linear Algebra, it previews a progression of topics including:

• Matrices
• Types of matrices
• Rank
• Row-reduced echelon form
• Systems of linear equations
• LU decomposition

The purpose of the series is to start from fundamentals and build toward high-level / GATE-level questions, emphasizing the correct approach to reduce time and errors.

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Detailed concepts covered about matrices

1) What is a matrix?

A matrix is formed by arranging numbers in rows and columns inside brackets.

• A 3×3 matrix has 3 rows and 3 columns (i.e., 9 entries).
• A 2×3 matrix has 2 rows and 3 columns.

2) How to denote elements of a matrix

• Matrices are typically named using a capital letter, e.g., A, B.
• Individual elements are denoted using a lowercase letter with indices, e.g., A(i, j) (often written as A₍i,j₎).

Examples:

• A₍1,1₎ = element in row 1, column 1
• A₍1,2₎ = element in row 1, column 2
• A₍2,3₎ = element in row 2, column 3

Similarly for B₍i,j₎.

3) Order (size/dimensions) of a matrix

The order of a matrix is:

• (number of rows) × (number of columns)
• Expressed as m×n (rows × columns)

Examples:

• For a 3×3 matrix: order is 3×3
• For a 2×3 matrix: order is 2×3

This determines the matrix’s structure (e.g., 5×5 means 5 rows and 5 columns).

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Matrix operations (conditions and rules)

4) Addition (and subtraction)

Matrix addition A + B is possible only if the matrices have the same order.

• Subtraction A − B also requires the same order.
• Rule: add/subtract entry-wise.

If corresponding entries are aᵢⱼ and bᵢⱼ, then:

• (A + B)ᵢⱼ = aᵢⱼ + bᵢⱼ

5) Scalar/commutativity note

• Addition is commutative:
  • A + B = B + A
• Subtraction is not commutative:
  • A − B ≠ B − A

6) Product of two matrices (AB)

Let:

• A be of order m×n
• B be of order p×q

Condition for AB to exist:

• The number of columns of A must equal the number of rows of B
  • i.e., n = p

Resulting order:

• AB has order m×q

How to compute the product:

Each entry of AB is computed as a dot product:

• Entry in row i of A and column j of B = (row i elements of A) · (column j elements of B)

The process includes:

• multiplying the first row of A with the first column of B
• multiplying the second row of A with each column of B
• and so on.

7) Commutativity vs associativity of matrix multiplication

• Matrix multiplication is not commutative:
  • AB may not equal BA
• Matrix multiplication is associative:
  • (AB)C = A(BC)
  • Changing grouping does not change the result.

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Transpose of a matrix

8) Definition

The transpose of matrix A is denoted Aᵗ.

• Rule: interchange rows and columns
• Order change:
  • If A is m×n, then Aᵗ is n×m

9) Properties of transpose

The video lists the following properties:

• Transpose of transpose:
  • (Aᵗ)ᵗ = A
• Transpose of sum:
  • (A + B)ᵗ = Aᵗ + Bᵗ
• Scalar with transpose:
  • (rA)ᵗ = rAᵗ
• Transpose of product:
  • (AB)ᵗ = BᵗAᵗ
• Multiple matrices:
  • (A₁A₂…Aₖ)ᵗ = Aₖᵗ … A₂ᵗA₁ᵗ
• Note referenced: relationship between inverse and transpose is mentioned but deferred for later videos.

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Types of matrices (definitions + key properties)

10) Row matrix

A matrix with only one row.

• Shape: 1×m

11) Column matrix

A matrix with only one column.

• Shape: n×1

12) Null / zero matrix

All entries are 0.

• Shape: can be m×n (any dimensions)

13) Square matrix

Rows = columns.

• Shape: n×n

14) Diagonal matrix

Must be square.

• All non-diagonal entries are zero
• Only diagonal entries may be non-zero

15) Scalar matrix

A diagonal matrix where all diagonal entries equal the same scalar (e.g., λ).

• Form: λ on the diagonal and 0 elsewhere

16) Identity matrix

A square matrix with:

• 1 on the diagonal
• 0 elsewhere

Property (for compatible matrices A):

• IA = A
• AI = A

17) Trace of a matrix

Defined only for square matrices.

• Trace = sum of diagonal entries

Example:

• If a 3×3 matrix has diagonal entries 1, 5, 9, then:
  • tr(A) = 1 + 5 + 9 = 15

Trace properties mentioned:

• Scalar multiple:
  • tr(cA) = c · tr(A)
• Additive:
  • tr(A ± B) = tr(A) ± tr(B)
• Cyclic property:
  • tr(AB) = tr(BA)
• Transpose doesn’t change trace:
  • tr(A) = tr(Aᵗ)
• Special values:
  • tr(I) = n
  • tr(null matrix) = 0
• Important restriction:
  • Do not confuse:
    • tr(AB) ≠ tr(A) · tr(B)
  • But tr(A + B) = tr(A) + tr(B)

18) Triangular matrices

• Upper triangular: entries below the diagonal are 0
• Lower triangular: entries above the diagonal are 0

Diagnostic tip:

• Zeros may appear on the diagonal; that’s fine—the key is whether the required side of the diagonal is zero.

Special cases:

• Null matrix is both upper and lower triangular.
• Diagonal matrix is both upper and lower triangular.

Transpose relationship:

• transpose(lower triangular) = upper triangular
• transpose(upper triangular) = lower triangular

19) Minimum number of zeros in a triangular matrix

For an n×n triangular matrix, the minimum number of forced zeros is:

• n(n − 1) / 2

Examples:

• For 3×3: 3(2)/2 = 3
• For 4×4: 4(3)/2 = 6

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Methodology / learning approach emphasized

• Learn fundamentals first, then apply them to GATE-level questions.
• Focus on the correct approach to reduce:
  • time spent
  • probability of errors
• For recorded videos:
  • when a question appears, pause and solve it yourself
  • then move to the provided solution
• Make and maintain notes:
  • write alongside the instructor
  • use PDFs/links provided in the video description
• Mentions staying consistent for GATE 2026 and GATE 2027.

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Speakers / sources featured

• No individual speaker name is provided in the subtitles.
• Source/Channel: Geeks for Geeks (and Great Computer Science and Data Science as part of the branding).