What defines a square matrix, and how is it denoted?

A square matrix has an equal number of rows and columns, denoted as Bₙ × ₙ.

Fill in the blank: A matrix with only one column and any number of rows is called a ___ matrix.

Column

True or False: A singular matrix has a determinant equal to zero.

True

What is the condition for a matrix to be classified as orthogonal?

A matrix B is orthogonal if B × Bᵀ = I, where I is the identity matrix.

What distinguishes a symmetric matrix from a skew-symmetric matrix?

A symmetric matrix satisfies Aᵀ = A, while a skew-symmetric matrix satisfies Aᵀ = -A.

What is the determinant of a zero matrix?

The determinant of a zero matrix is zero.

Which type of matrix represents probabilities, ensuring that each row sums to one?

A stochastic matrix

True or False: The inverse of a non-square matrix exists.

False; only square matrices have inverses.

How can you tell if a matrix is consistent when solving linear equations?

A system of equations is consistent if the rank of the augmented matrix equals the rank of the coefficient matrix.

What is the relationship between the determinant of a matrix and its transpose?

The determinant of a matrix is the same as the determinant of its transpose.

Fill in the blank: For a matrix to be non-singular, its determinant must be ___ zero.

Not