Let matrix B be the adjoint of a square matrix A, l be the identify matrix of same order as A. If k (≠ 0) is the determinate of the matrix A, then what is AB equal to?
kl
Understanding Matrix Adjoint and Determinant
The question asks for the value of the product of a square matrix A and its adjoint matrix B, given that the determinant of A is k and l is the identity matrix of the same order as A.
Key Concepts
- Square Matrix: A matrix with the same number of rows and columns.
- Adjoint of a Matrix (adj(A)): The transpose of the cofactor matrix of A. In this problem, matrix B is given as the adjoint of A, so \(B = \text{adj}(A)\).
- Determinant of a Matrix (\(\det(A)\) or \(|A|\)): A scalar value calculated from the elements of a square matrix. It is given that \(\det(A) = k\). Since \(k \ne 0\), matrix A is non-singular.
- Identity Matrix (I or l): A square matrix with ones on the main diagonal and zeros elsewhere. When multiplied by any matrix of the same order, it leaves the matrix unchanged. In this problem, the identity matrix is denoted by l.
Relationship between Matrix, Adjoint, Determinant, and Identity Matrix
There is a fundamental property relating a square matrix A, its adjoint adj(A), its determinant \(\det(A)\), and the identity matrix I of the same order. This property is given by the equation:
\[ A \cdot \text{adj}(A) = \det(A) \cdot I \]
Applying the Property to the Given Problem
We are given:
- Matrix A
- Matrix B is the adjoint of A, so \(B = \text{adj}(A)\).
- The determinant of matrix A is k, so \(\det(A) = k\).
- The identity matrix is l, so \(I = l\).
Substituting these given values into the property \(A \cdot \text{adj}(A) = \det(A) \cdot I\), we get:
\[ A \cdot B = k \cdot l \]
Thus, the product AB is equal to kl.
Conclusion
Based on the fundamental property of matrices, the product of a square matrix A and its adjoint B (where B = adj(A)) is equal to the determinant of A multiplied by the identity matrix of the same order. Given \(\det(A) = k\) and the identity matrix is l, we find that \(AB = kl\).
Matching with Options
Comparing our result \(kl\) with the given options:
- l
- kl
- k2l
- (1/k)l
Our result matches option 2.
Revision Table: Matrix Properties
| Property | Description | Formula |
|---|---|---|
| Matrix times its Adjoint | The product of a square matrix and its adjoint is the determinant times the identity matrix. | \(A \cdot \text{adj}(A) = \text{adj}(A) \cdot A = \det(A) \cdot I\) |
| Determinant of Adjoint | The determinant of the adjoint of an n x n matrix A. | \(\det(\text{adj}(A)) = (\det(A))^{n-1}\) |
| Adjoint of Adjoint | The adjoint of the adjoint of an n x n matrix A. | \(\text{adj}(\text{adj}(A)) = (\det(A))^{n-2} \cdot A\) |
| Inverse of a Matrix | If \(\det(A) \ne 0\), the inverse of A exists. | \(A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\) |
Additional Information: Related Matrix Concepts
- Singular Matrix: A square matrix whose determinant is zero (\(\det(A) = 0\)). A singular matrix does not have an inverse. For a singular matrix A, \(A \cdot \text{adj}(A) = 0\), where 0 is the zero matrix.
- Non-Singular Matrix: A square matrix whose determinant is non-zero (\(\det(A) \ne 0\)). A non-singular matrix has a unique inverse.
- Cofactor Matrix: A matrix where each element \((C_{ij})\) is the cofactor of the corresponding element \((a_{ij})\) in the original matrix. The adjoint is the transpose of this matrix.
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