CHAPTER 3: MATRICES
Class 12 CBSE Mathematics Core High-Efficiency Formula Sheet
1. Matrix Definitions & Algebra
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called
the elements or the entries of the matrix.
A = [aij]m × n
Where m represents rows and n represents columns.
2. Special Types of Matrices
• Square Matrix: A matrix in which the number of rows is equal to the number of columns (m = n).
• Diagonal Matrix: A square matrix in which all non-diagonal elements are strictly zero.
• Scalar Matrix: A diagonal matrix in which all diagonal elements are equal to a constant scalar k.
• Identity Matrix: A diagonal matrix in which all diagonal elements are exactly 1, denoted by I.
3. Transpose & Symmetry Operations
The transpose of a matrix A, denoted as A' or AT, is obtained by interchanging its rows and columns.
Properties of Transpose:
• (A')' = A
• (kA)' = kA'
• (A + B)' = A' + B'
• (AB)' = B'A' (Reversal Law)
Symmetric Matrix: A square matrix satisfies A' = A.
Skew-Symmetric Matrix: A square matrix satisfies A' = -A. Note: All diagonal elements of a skew-
symmetric matrix must be zero (aii = 0).
Theorem: Any square matrix A can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix:
A = ½(A + A') + ½(A - A')
The Power of 'No': To be a topper, you have to say 'no' to the things that don't matter right now—extra sleep,
mindless scrolling, or hanging out. The result day will prove it was worth it. ego never accepts defeat.
Chapter 3 Notes • Page 1
Class 12 CBSE Mathematics Core High-Efficiency Formula Sheet
1. Matrix Definitions & Algebra
A matrix is an ordered rectangular array of numbers or functions. The numbers or functions are called
the elements or the entries of the matrix.
A = [aij]m × n
Where m represents rows and n represents columns.
2. Special Types of Matrices
• Square Matrix: A matrix in which the number of rows is equal to the number of columns (m = n).
• Diagonal Matrix: A square matrix in which all non-diagonal elements are strictly zero.
• Scalar Matrix: A diagonal matrix in which all diagonal elements are equal to a constant scalar k.
• Identity Matrix: A diagonal matrix in which all diagonal elements are exactly 1, denoted by I.
3. Transpose & Symmetry Operations
The transpose of a matrix A, denoted as A' or AT, is obtained by interchanging its rows and columns.
Properties of Transpose:
• (A')' = A
• (kA)' = kA'
• (A + B)' = A' + B'
• (AB)' = B'A' (Reversal Law)
Symmetric Matrix: A square matrix satisfies A' = A.
Skew-Symmetric Matrix: A square matrix satisfies A' = -A. Note: All diagonal elements of a skew-
symmetric matrix must be zero (aii = 0).
Theorem: Any square matrix A can be uniquely expressed as the sum of a symmetric and a skew-
symmetric matrix:
A = ½(A + A') + ½(A - A')
The Power of 'No': To be a topper, you have to say 'no' to the things that don't matter right now—extra sleep,
mindless scrolling, or hanging out. The result day will prove it was worth it. ego never accepts defeat.
Chapter 3 Notes • Page 1
