Identify which one of the following statements is CORRECT:
A -1 is equal to Adj (A) if the value of the determinant A is unity
Analyzing Correct Statements about Matrices and Determinants
Let's analyze each statement provided in the options to determine which one is correct based on the properties of matrices and determinants.
Statement 1: Matrix Inverse and Adjoint Relationship
The first statement says that \(A^{-1}\) is equal to \(\text{Adj}(A)\) if the value of the determinant of \(A\) is unity. Let's recall the formula for the inverse of a square matrix \(A\):
\[A^{-1} = \frac{1}{\text{det}(A)} \text{Adj}(A)\]
Here, \(\text{det}(A)\) is the determinant of matrix \(A\), and \(\text{Adj}(A)\) is the adjoint of matrix \(A\). The inverse exists only if \(\text{det}(A) \neq 0\). If the value of the determinant \(A\) is unity, which means \(\text{det}(A) = 1\), the formula becomes:
\[A^{-1} = \frac{1}{1} \text{Adj}(A)\]
\[A^{-1} = \text{Adj}(A)\]
So, if the determinant of matrix \(A\) is 1, its inverse \(A^{-1}\) is indeed equal to its adjoint \(\text{Adj}(A)\). This statement appears to be correct.
Statement 2: Determinant Property - Column Interchange
The second statement claims that if two columns of a determinant are interchanged, the sign of the determinant does not alter. This is incorrect. A fundamental property of determinants states that if any two rows or any two columns of a determinant are interchanged, the sign of the determinant changes (it gets multiplied by -1). For example, if \(\text{det}(A) = k\), interchanging two columns results in a new determinant with a value of \(-k\). Therefore, this statement is incorrect.
Statement 3: Determinant Property - Transpose
The third statement asserts that the value of a determinant changes if the rows and columns are interchanged. Interchanging rows and columns of a matrix is the process of taking its transpose. A key property of determinants is that the determinant of a matrix is equal to the determinant of its transpose, i.e., \(\text{det}(A) = \text{det}(A^T)\). This means interchanging rows and columns does not change the value of the determinant. Therefore, this statement is incorrect.
Statement 4: Matrix Operations - Division
The fourth statement suggests that the law of division is allowed in matrix operations. Standard matrix algebra defines addition, subtraction, and multiplication. There is no standard operation of "division" defined for matrices in the same way it exists for numbers. While multiplying by the inverse (\(A^{-1}\)) can be thought of as analogous to division in some contexts (like solving linear equations \(AX = B\) by multiplying by \(A^{-1}\) to get \(X = A^{-1}B\)), direct matrix division \(A/B\) or \(A \div B\) is not a defined operation. Therefore, this statement is incorrect.
Identifying the Correct Statement on Matrix Properties
Based on the analysis of each statement, only the first statement accurately describes a property related to matrices and determinants. The relationship between the matrix inverse, the adjoint, and the determinant is correctly stated under the specific condition where the determinant is unity.
Revision Table: Key Matrix and Determinant Concepts
| Concept | Description / Property |
|---|---|
| Matrix Inverse (\(A^{-1}\)) | Exists for a square matrix \(A\) if \(\text{det}(A) \neq 0\). Formula: \(A^{-1} = \frac{1}{\text{det}(A)} \text{Adj}(A)\) |
| Adjoint of a Matrix (\(\text{Adj}(A)\)) | The transpose of the cofactor matrix of \(A\). |
| Determinant Property (Column/Row Swap) | Interchanging two rows or columns changes the sign of the determinant. |
| Determinant Property (Transpose) | The determinant of a matrix is equal to the determinant of its transpose (\(\text{det}(A) = \text{det}(A^T)\)). |
| Matrix Division | Not a standard defined operation in matrix algebra. Multiplication by the inverse serves a similar purpose in some applications. |
Additional Information on Matrix Operations
Understanding the basic operations and properties of matrices and determinants is crucial in linear algebra. Matrix addition and subtraction are defined element-wise for matrices of the same dimensions. Matrix multiplication is defined for matrices \(A\) (m x n) and \(B\) (n x p), resulting in a matrix \(AB\) (m x p). Matrix multiplication is not commutative in general (\(AB \neq BA\)). Determinants are scalar values associated with square matrices, providing important information like whether the matrix is invertible (\(\text{det}(A) \neq 0\)). The adjoint matrix is used in the calculation of the inverse matrix and also relates to Cramer's rule for solving systems of linear equations.
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