CHAPTER 4: DETERMINANTS
Class 12 CBSE Mathematics Core High-Efficiency Formula Sheet
1. Evaluation and Scalar Scaling
To every square matrix A = [aij] of order n, we can associate a unique number called the determinant of
A, denoted by |A|.
Critical Scaling Property: If A is a square matrix of order n × n, scaling the entire matrix by a scalar
factor k scales the determinant exponentially:
|kA| = kn|A|
2. Core Operational Properties
• Product Property: |AB| = |A||B| where A and B are square matrices of the same order.
• Singular Matrix: A square matrix A is called singular if |A| = 0. If |A| ≠ 0, it is non-singular.
3. Adjoint and Inverse Identities
The adjoint of a square matrix A is the transpose of the cofactor matrix.
• A · (adj A) = (adj A) · A = |A|I
• |adj A| = |A|n-1
• Matrix Inverse: A-1 = (1 / |A|) · adj A (Only holds when |A| ≠ 0)
4. Solving Systems of Linear Equations
For a system modeled as AX = B:
• If |A| ≠ 0, the system is consistent and possesses a unique solution given by: X = A-1B.
• If |A| = 0 and (adj A)B ≠ O, the system is inconsistent and has no solution.
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 4 Notes • Page 1
Class 12 CBSE Mathematics Core High-Efficiency Formula Sheet
1. Evaluation and Scalar Scaling
To every square matrix A = [aij] of order n, we can associate a unique number called the determinant of
A, denoted by |A|.
Critical Scaling Property: If A is a square matrix of order n × n, scaling the entire matrix by a scalar
factor k scales the determinant exponentially:
|kA| = kn|A|
2. Core Operational Properties
• Product Property: |AB| = |A||B| where A and B are square matrices of the same order.
• Singular Matrix: A square matrix A is called singular if |A| = 0. If |A| ≠ 0, it is non-singular.
3. Adjoint and Inverse Identities
The adjoint of a square matrix A is the transpose of the cofactor matrix.
• A · (adj A) = (adj A) · A = |A|I
• |adj A| = |A|n-1
• Matrix Inverse: A-1 = (1 / |A|) · adj A (Only holds when |A| ≠ 0)
4. Solving Systems of Linear Equations
For a system modeled as AX = B:
• If |A| ≠ 0, the system is consistent and possesses a unique solution given by: X = A-1B.
• If |A| = 0 and (adj A)B ≠ O, the system is inconsistent and has no solution.
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 4 Notes • Page 1
