Question

Let \(A\) and \(B\) be invertible \(n \times n\) matrices. Show that \(\operatorname{det} A=\operatorname{det} B\) if and only if \(A=U B\) where \(U\) is a matrix with \(\operatorname{det} U=1\).

Short answer

\(\det A = \det B\) if and only if \(A = UB\) with \(\det U = 1\).

Step-by-step solution

Understanding Determinant Properties

Recall that the determinant of a matrix product equals the product of their determinants, i.e., \( \operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B) \). Also remember that \( \operatorname{det}(A^{-1}) = \frac{1}{\operatorname{det}(A)} \) for an invertible matrix \( A \). These properties will be crucial in proving the statement.

Assume \( A = UB \) and Prove \( \operatorname{det} A = \operatorname{det} B \)

Assume \( A = UB \) with \( \operatorname{det}(U) = 1 \). Then, the determinant of \( A \) becomes \( \operatorname{det}(A) = \operatorname{det}(UB) = \operatorname{det}(U) \times \operatorname{det}(B) = 1 \times \operatorname{det}(B) = \operatorname{det}(B) \). This proves \( \operatorname{det} A = \operatorname{det} B \).

Assume \( \operatorname{det} A = \operatorname{det} B \) and Prove \( A = UB \)

Assume \( \operatorname{det} A = \operatorname{det} B \). We need to show there exists a matrix \( U \) such that \( \operatorname{det}(U) = 1 \) and \( A = UB \). Since both matrices are invertible, consider \( U = AB^{-1} \). We then have \( A = UB = (AB^{-1})B \). Therefore, \( U = AB^{-1} \). Now, the determinant of \( U \) is \( \operatorname{det}(U) = \operatorname{det}(AB^{-1}) = \frac{\operatorname{det}(A)}{\operatorname{det}(B)} = \frac{\operatorname{det}(B)}{\operatorname{det}(B)} = 1 \).

Conclusion

From steps 2 and 3, we have shown that if \( A = UB \) with \( \operatorname{det}(U) = 1 \), then \( \operatorname{det} A = \operatorname{det} B \). Conversely, if \( \operatorname{det} A = \operatorname{det} B \), there exists \( U = AB^{-1} \) with \( \operatorname{det}(U) = 1 \) such that \( A = UB \). Hence, the statement \( \operatorname{det}(A) = \operatorname{det}(B) \) if and only if \( A = UB \) where \( U \) has \( \operatorname{det}(U) = 1 \) is proven.

Key concepts

matrix multiplication

Matrix multiplication is a fundamental operation in linear algebra employed to combine matrices in a way that corresponds to applying one transformation after another. When multiplying two matrices, the number of columns in the first matrix must match the number of rows in the second. The resulting matrix will have dimensions corresponding to the rows of the first matrix and the columns of the second.

Each element of the resulting matrix is computed by taking the dot product of the corresponding row from the first matrix and the column from the second matrix. For example, if you are multiplying matrix \( A \) and matrix \( B \), the element in the first row, first column of the resulting matrix is obtained by multiplying each element of the first row of \( A \) with each corresponding element of the first column of \( B \), and summing the results.

• Order matters in multiplication; \( AB \) might not equal \( BA \).
• Matrix multiplication is associative, meaning \( (AB)C = A(BC) \).
• Matrix multiplication distributes over addition: \( A(B + C) = AB + AC \).

These properties make matrix multiplication an essential tool when dealing with transformations and changes of basis in vector spaces.

invertible matrices

An invertible matrix, also known as a non-singular matrix, is a square matrix that possesses an inverse. The inverse of a matrix \( A \) is another matrix \( A^{-1} \) such that when \( A \) is multiplied by \( A^{-1} \), the identity matrix is obtained: \( AA^{-1} = I \) and \( A^{-1}A = I \). An identity matrix is a diagonal matrix with ones on the diagonal and zeros elsewhere, effectively acting as the ‘multiplicative identity’ in matrix operations.

For a matrix to be invertible, it must be square (having the same number of rows and columns) and its determinant must be non-zero. This ensures the rows and columns of the matrix are linearly independent, facilitating the existence of the inverse.

• If a matrix is invertible, there exist methods such as the Gaussian elimination, adjoint method, or using determinants to compute its inverse.
• An invertible matrix has full rank, meaning all its rows and columns are independent.

Understanding invertible matrices is crucial for solving systems of linear equations, as invertibility directly impacts unique solutions.

matrix determinant properties

The determinant of a matrix is a scalar value that can provide essential insights into the properties of the matrix. Primarily, it indicates whether a matrix is invertible; a matrix is invertible only if its determinant is not zero.

There are several key properties of the determinant:

• For any matrices \( A \) and \( B \), \( \operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B) \).
• The determinant of a triangular matrix (upper or lower) is the product of its diagonal elements.
• Swapping two rows of a matrix changes the sign of its determinant.
• If a row or a column of a matrix is scaled by a factor, the determinant is multiplied by that factor.
• The determinant of matrix \( A^{-1} \) is \( 1/\operatorname{det}(A) \).

These properties come handy when analyzing transformations characterized by matrices, as they help in determining geometrical features such as volume scaling and checking whether transformation matrices affect orientation.