What is a square matrix and how is its order defined?

A square matrix is a matrix that has an equal number of rows and columns. The order of a square matrix is defined as n × n, where n is the number of rows (or columns).

The trace of a square matrix A is given by the formula ___

Trace(A) = Σ aᵢᵢ, where aᵢᵢ represents the elements on the leading diagonal of the matrix A.

True or False: A diagonal matrix is a square matrix where all elements outside the leading diagonal are zero.

True. A diagonal matrix has all non-diagonal elements equal to zero.

What defines a scalar matrix?

A scalar matrix is a diagonal matrix in which all the elements in the leading diagonal are equal to the same scalar value K.

What is the identity matrix and how is it denoted?

The identity matrix is a square matrix in which all the elements on the leading diagonal are 1 and all other elements are 0. It is denoted by Iₙ, where n is the order of the matrix.

Define the transpose of a matrix A and its properties.

The transpose of a matrix A, denoted A', is obtained by interchanging its rows and columns. Properties include: (A')' = A, (A + B)' = A' + B', and (kA)' = kA' for any scalar k.

A matrix A is called skew-symmetric if ___

A matrix A is called skew-symmetric if aᵢⱼ = -aⱼᵢ for all i and j, which implies that all diagonal elements are zero.

What is a non-singular matrix?

A non-singular matrix is a square matrix whose determinant is non-zero, indicating that it has an inverse.

Define the cofactor of an element aᵢⱼ in a determinant.

The cofactor of an element aᵢⱼ is defined as Cᵢⱼ = (-1)^(i+j) Mᵢⱼ, where Mᵢⱼ is the minor obtained by deleting the ith row and jth column.

What is Cramer's Rule and when is it applicable?

Cramer's Rule provides a way to solve a system of linear equations using determinants. It is applicable when the system has a unique solution, which occurs if the determinant of the coefficient matrix is non-zero.

The product of two matrices A and B is defined only when ___

The product of two matrices A (of size m × n) and B (of size n × p) is defined when the number of columns in A is equal to the number of rows in B.

What is the determinant of a 2 × 2 matrix given by |a b; c d|?

The determinant of a 2 × 2 matrix is calculated as |A| = ad - bc.

True or False: The multiplication of matrices is commutative.

False. In general, the multiplication of matrices is not commutative; that is, AB ≠ BA.

Explain the significance of the null matrix.

A null matrix, denoted by 0, is a matrix where all elements are zero. It acts as the additive identity in matrix addition.

If A is a square matrix and |A| = 0, then A is called ___

A is called a singular matrix.