A non-singular matrix is a square matrix whose determinant is not equal to zero. Such matrices are invertible, meaning they have an inverse.
Singular Matrix: A square matrix with determinant = 0. Non-Singular Matrix: A square matrix with determinant ≠ 0.
The key theorem states: A square matrix A of order n is non-singular if and only if its rank is equal to n. If det(A)≠0, all rows (or columns) are linearly independent, so rank = n. Conversely, if rank = nn, then det(A)≠0, and the matrix is non-singular.
By calculating its determinant. If it is not zero, the matrix is non-singular.
Yes, every invertible matrix is non-singular.
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Non-Singular Matrix
In linear algebra, matrices are the most essential part of higher math, especially when getting ready for the JEE. Non-singular matrices are necessary for solving equations, figuring out if something can be inverted, and looking at systems of linear equations.
A non-singular matrix is one whose determinant is not zero. This property sets it apart from a singular matrix, which has a determinant that equals zero. To do well on competitive tests like the JEE, you need to know the non-singular matrix formula, its properties, and some examples.
1.0What Is a Non-Singular Matrix?
A non-singular matrix is a square matrix (the number of rows equals the number of columns) that has a non-zero determinant. In mathematical terms, if ( A ) is a square matrix and ( det(A)=0 ), then ( A ) is non-singular.
Condition: Only square matrices (order n×n) can be non-singular.
Key Insight: A non-singular matrix always has an inverse, which is why it is sometimes called an invertible matrix.
2.0Non-Singular Matrix Formula
The most crucial formula related to non-singular matrices is the inverse formula.
If A is a non-singular matrix of order n, its inverse is given by:
A−1=det(A)Adj(A)
Where:
Adj(A) = Adjoint of the matrix A
det(A) ≠ 0
This formula highlights the fact that an inverse exists only when the matrix is non-singular.
3.0How To Find A Non-Singular Matrix?
The determinant test is the main way we find out if a matrix is non-singular. This process is simple and follows a set pattern because the determinant is what matters. Let's take it apart:
Step 1: Check if the Matrix is Square
Only square matrices (matrices with an equal number of rows and columns) can be classified as singular or non-singular.
If the matrix is not square, the concept of non-singular does not apply.
Step 2: Calculate the Determinant
For a 2 × 2 matrix: A=[acbd]det(A)=ad−bc
For a 3 × 3 matrix: A=adgbehcfi= det(A)=a(ei−fh)−b(di−fg)+c(dh−eg)
For higher-order matrices, determinants are calculated using expansion by minors or row-reduction methods.
Step 3: Apply the Determinant Condition
If det(A)=0, then the matrix is singular.
If det(A)≠0, then the matrix is non-singular.
Step 4: Verify with Rank (Optional for JEE-level cross-check)
A square matrix of order n is non-singular if its rank = n.
This means all rows (or columns) are linearly independent.
Step 5: Check for Inverse (Alternative Method)
If you can compute an inverse for the matrix using: A−1=det(A)Adj(A), then A must be non-singular, since this formula is valid only when det(A)≠0.
4.0Rules For Row and Column Operations of a Determinant
While solving determinants, especially for larger matrices, row and column operations simplify the process. But certain rules must be followed carefully:
Swapping Rows or Columns: If any two rows (or columns) are interchanged, the sign of the determinant changes.
det(A)→−det(A)
Identical Rows or Columns: The determinant is zero when two rows (or columns) are either the same, or proportional.
Row/Column Multiplication: When you multiply a row or column (of any matrix) by a scalar k, the determinant is multiplied by k (the same scalar).
Row/Column Addition Rule: Adding a multiple of one row to another (or a column to another) does not affect the value of the determinant. This property will be used to simplify matrices before finding a determinant.
Zero Row or Column: If any row (or column) is completely zero, the determinant is zero and the matrix is singular.
5.0Properties of Non-Singular Matrix
Understanding the non-singular matrix properties is vital for JEE and advanced mathematics: