Understanding the Determinant of \(B^{-1}AB\)

The question asks for the value of the determinant of the matrix product \(B^{-1}AB\), given that \(B\) is a non-singular matrix and \(A\) is a square matrix. To solve this, we need to use some important properties of determinants.

Key Determinant Properties

There are several fundamental properties of determinants that are useful in matrix algebra. The properties relevant to this problem are:

• The determinant of a product of matrices is the product of their determinants. For square matrices \(X\) and \(Y\), \(\det(XY) = \det(X) \det(Y)\). This property can be extended to a product of three or more matrices, like \(\det(XYZ) = \det(X) \det(Y) \det(Z)\).
• If a matrix \(B\) is non-singular (or invertible), its determinant \(\det(B)\) is not equal to zero. Its inverse \(B^{-1}\) exists, and the determinant of the inverse is the reciprocal of the determinant of the original matrix: \(\det(B^{-1}) = \frac{1}{\det(B)}\).

Calculating \(\det(B^{-1}AB)\)

Now let's apply these properties to find the determinant of the given matrix product \(B^{-1}AB\).

We have the expression \(\det(B^{-1}AB)\). Using the property for the determinant of a product of three matrices, we can write:

\(\det(B^{-1}AB) = \det(B^{-1}) \det(A) \det(B)\)

Since \(B\) is a non-singular matrix, its inverse \(B^{-1}\) exists, and we know that \(\det(B^{-1}) = \frac{1}{\det(B)}\). Substituting this into the equation:

\(\det(B^{-1}AB) = \left(\frac{1}{\det(B)}\right) \det(A) \det(B)\)

Now, we can rearrange the terms and cancel out \(\det(B)\), as \(\det(B) \neq 0\) because \(B\) is non-singular:

\(\det(B^{-1}AB) = \det(A) \left(\frac{\det(B)}{\det(B)}\right)\)

\(\det(B^{-1}AB) = \det(A) \times 1\)

\(\det(B^{-1}AB) = \det(A)\)

Thus, the value of \(\det(B^{-1}AB)\) is equal to the determinant of matrix \(A\).

Conclusion

Based on the properties of determinants and non-singular matrices, the determinant of the matrix product \(B^{-1}AB\) simplifies to the determinant of matrix \(A\).

Let's review the options provided:

• Option 1: \(\det(B)\)
• Option 2: \(\det(A)\)
• Option 3: \(\det(B^{-1})\)
• Option 4: \(\det(A^{-1})\)

Our calculated result matches Option 2.

Revision Table: Matrix Determinant Properties

Additional Information: Similar Matrices and Determinants

The expression \(B^{-1}AB\) is important in linear algebra because it represents a matrix \(A\) transformed by an invertible matrix \(B\). Matrices of the form \(B^{-1}AB\) and \(A\) are called similar matrices.

An interesting property is that similar matrices have the same determinant. Our calculation confirms this property:

\(\det(B^{-1}AB) = \det(A)\)

This means that the determinant is invariant under similarity transformations. Similar matrices share many properties, including their determinant, trace, eigenvalues, and characteristic polynomial. This concept is fundamental in understanding how linear transformations behave under a change of basis.

Understanding the properties of determinants and matrix operations, especially involving non-singular matrices and inverses, is crucial for solving problems in linear algebra.