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How to Calculate the Adjoint of a matrix? Video Lecture | Mathematics (Maths) for JEE Main & Advanced
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FAQs on How to Calculate the Adjoint of a matrix? Video Lecture - Mathematics (Maths) for JEE Main & Advanced
| 1. How do I calculate the adjoint of a matrix? |  |
To calculate the adjoint of a matrix, follow these steps:
1. Find the cofactor of each element in the matrix. The cofactor of an element is the determinant of the submatrix obtained by removing the row and column containing that element.
2. Create a new matrix called the cofactor matrix, where each element is replaced by its corresponding cofactor.
3. Transpose the cofactor matrix by interchanging its rows with columns to obtain the adjoint matrix.
For example, let's say we have a 3x3 matrix A:
A = | a11 a12 a13 |
| a21 a22 a23 |
| a31 a32 a33 |
The adjoint of A, denoted as adj(A), can be calculated by finding the cofactor matrix and then transposing it.
| 2. What is the purpose of calculating the adjoint of a matrix? | |
The adjoint of a matrix is often used to find the inverse of a matrix. If a matrix A is invertible (non-singular), then its inverse A^-1 can be obtained by dividing the adjoint of A by the determinant of A. In other words, A^-1 = adj(A) / det(A). The adjoint is also useful in solving systems of linear equations and in various applications of linear algebra.
| 3. Can the adjoint of a matrix be equal to the original matrix? | |
No, the adjoint of a matrix is not equal to the original matrix unless the original matrix is an orthogonal matrix. An orthogonal matrix is a square matrix where the transpose of the matrix is equal to its inverse. In such cases, the adjoint and the original matrix are the same.
| 4. Is the adjoint of a matrix always a square matrix? | |
Yes, the adjoint of a matrix is always a square matrix, regardless of the size or dimensions of the original matrix. If the original matrix is an m x n matrix, then its adjoint will be an n x m matrix.
| 5. Can the adjoint of a matrix be negative? | |
No, the adjoint of a matrix cannot be negative. The adjoint matrix contains the cofactors of the original matrix, and a cofactor is a determinant. Determinants are always non-negative values, so the adjoint matrix will also have non-negative elements.
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Nov 21, 2024 Last updated |
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