Product of a Matrix and its Transpose

The product of a matrix and its transpose results in an identity matrix. This property holds true for square matrices, where the number of rows is equal to the number of columns. Let's explore this phenomenon and understand the reasoning behind it.

Definition of a Transpose

The transpose of a matrix is obtained by interchanging its rows and columns. If matrix A has dimensions m x n, then its transpose, denoted as A^T, will have dimensions n x m. The element in the i-th row and j-th column of the transpose will be the element in the j-th row and i-th column of the original matrix.

Product of a Matrix and its Transpose

When we multiply a matrix by its transpose, the resulting matrix will have dimensions equal to the number of rows in the original matrix. If matrix A has dimensions m x n, then the product A * A^T will have dimensions m x m.

Identity Matrix

An identity matrix is a square matrix where all the elements on the main diagonal are 1, and all other elements are 0. It is denoted as I, and its size is specified by a subscript. For example, I2 represents a 2x2 identity matrix, I3 represents a 3x3 identity matrix, and so on.

Product of a Matrix and its Transpose is an Identity Matrix

When we take the product of a matrix A and its transpose A^T, we obtain a square matrix with dimensions m x m. This resulting matrix will be an identity matrix if the original matrix A satisfies certain conditions, namely:
1. The original matrix A must be a square matrix (m x m).
2. The columns of matrix A must be orthogonal (perpendicular) to each other.
3. The columns of matrix A must be normalized (have unit length).

Determinant of the Matrix

The determinant of a matrix is a scalar value that represents certain properties of the matrix. In the case of the product of a matrix and its transpose being an identity matrix, the determinant of this matrix will be 1.

Explanation of the Determinant

The determinant of a matrix reflects its effect on the scaling of space. When the product of a matrix and its transpose is an identity matrix, it implies that the original matrix preserves the orientation and shape of vectors. In other words, the matrix does not stretch, shrink, or invert the vectors it operates on.

Since the determinant of the identity matrix is 1, the determinant of the matrix resulting from the product of a matrix and its transpose will also be 1. The determinant captures the effect of the matrix on the volume of space, and in this case, it confirms that the matrix does not alter the volume.

Conclusion

The product of a matrix and its transpose results in an identity matrix when certain conditions are met. The determinant of this matrix is 1, indicating that the original matrix does not alter the orientation, shape, or volume of the vectors it operates on.

always 1