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Adjugate Matrix Formula:
where \( C \) is the cofactor matrix of \( A \)
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The adjugate matrix (also called the adjoint matrix) of a square matrix is the transpose of its cofactor matrix. It is primarily used in calculating the inverse of a matrix and has applications in linear algebra and differential equations.
The adjugate matrix is calculated using the formula:
Where:
Steps:
Applications: The adjugate matrix is used in finding the inverse of a matrix (\( A^{-1} = \frac{\text{adj}(A)}{\det(A)} \)), solving systems of linear equations, and in various theorems in linear algebra.
Instructions: Enter your square matrix as comma-separated values with each row on a new line. The calculator will compute the adjugate matrix by first calculating the cofactor matrix and then transposing it.
Q1: What's the difference between adjugate and adjoint?
A: In modern usage, adjugate and adjoint mean the same thing for matrices. However, "adjoint" can sometimes refer to the conjugate transpose in other contexts.
Q2: Does every matrix have an adjugate?
A: Yes, every square matrix has an adjugate. However, only invertible matrices (those with non-zero determinant) have an adjugate that can be used to find their inverse.
Q3: What's the relationship between adjugate and determinant?
A: The adjugate is used in the formula for the inverse: \( A^{-1} = \frac{\text{adj}(A)}{\det(A)} \). For singular matrices (det=0), the adjugate still exists but can't be used to find an inverse.
Q4: How does the adjugate relate to eigenvalues?
A: The adjugate appears in the characteristic polynomial of a matrix and can be used to find eigenvectors corresponding to eigenvalue zero.
Q5: Can the adjugate be computed for non-square matrices?
A: No, the adjugate is only defined for square matrices. The concept doesn't extend to rectangular matrices.