A square matrix has a determinant not equal to zero state at

A square matrix has a determinant not equal to zero. state atleast three other properties that the matrix must have.

Solution

If the matrix is square, it is possible to deduce some of its properties by computing its determinant. For example, a square matrix has an inverse if and only if its determinant is not zero

A square matrix is a matrix with the same number of rows and columns. An n-by-n matrix is known as a square matrix of order n. Any two square matrices of the same order can be added and multiplied. The entries a_ii form the main diagonal of a square matrix. They lie on the imaginary line which runs from the top left corner to the bottom right corner of the matrix.

A square matrix A that is equal to its transpose, i.e., A = A^T, is a symmetric matrix. If instead, A was equal to the negative of its transpose, i.e., A = A^T, then A is a skew-symmetric matrix

A square matrix A is called invertible or non-singular if there exists a matrix B such that

AB = BA = I_n

The trace, tr(A) of a square matrix A is the sum of its diagonal entries. While matrix multiplication is not commutative as mentioned above, the trace of the product of two matrices is independent of the order of the factors:

tr(AB) = tr(BA).