Virtual Labs

Solution of system of linear equations using linear map equation

Let

A=\begin{bmatrix} 1 & 2 \\ -2 & 1 \end{bmatrix}

and

B=\begin{bmatrix} 1 & 2 | 4 \\ -2 & 1 | 1 \end{bmatrix}

. Then

a: RankA=2; RankB=1 Explanation

Explanation

b: RankA=1; RankB=2 Explanation

Explanation

c: RankA=2; RankB=2 Explanation

Explanation

d: RankA=1; RankB=1 Explanation

Explanation

Let

A=\begin{bmatrix} -1 & 1 \\ -2 & 2 \end{bmatrix}

and

B=\begin{bmatrix} -1 & 2 | 0 \\ -2 & -1 | 0 \end{bmatrix}

. Then the rankA and rankB are respectively

a: 2 & 1 Explanation

Explanation

b: 1 & 2 Explanation

Explanation

c: 2 & 2 Explanation

Explanation

d: 0 & 1 Explanation

Explanation

Let A be a matrix of order n×m and the B be a matrix of order n×(m+1) such that the first n column of B are same as that of A. Then

a: RankA≤RankB Explanation

Explanation

b: RankA=RankB Explanation

Explanation

c: RankA≥RankB Explanation

Explanation

d: RankA=m & RankB=m+1 Explanation

Explanation

Let

A=\begin{bmatrix} 1 & -1 & 0 \\ 0 & 1 & -1 \end{bmatrix}

and

B=\begin{bmatrix} -1 \\ -1 \\ -2 \end{bmatrix}

. Then A.B equals

\begin{bmatrix} 2 \\ 3 \end{bmatrix}

Explanation

\begin{bmatrix} -2 \\ 3 \\ 0 \end{bmatrix}

Explanation

\begin{bmatrix} 2 \\ 3 \end{bmatrix}

Explanation

d: Not defined Explanation

Explanation

Let

A=\begin{bmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{bmatrix}

a: A is invertible and A^2 is invertible Explanation

Explanation

b: A is not invertible and A^2 is invertible Explanation

Explanation

c: A is invertible and A^2 is not invertible Explanation

Explanation

d: A is not invertible and A^2 is also not invertible Explanation

Explanation

Let

A=\begin{bmatrix} 1 & 1 & 0 \\ 1 & -1 & 1 \\ 2 & 1 & 0 \end{bmatrix}

a: det(A)=2 and det(A^{-1}) does not exist Explanation

Explanation

b: det(A)=-1 and det(A^{-1}) exists Explanation

Explanation

c: det(A)=1and det(A^{-1}) exists Explanation

Explanation

d: det(A)=1 and det(A^{-1}) does not exist Explanation

Explanation

Let

A=\begin{bmatrix} 4 & -1 \\ 12 & -3 \end{bmatrix}

a: A is invertible and A=A^2 Explanation

Explanation

b: A is not invertible and A=A^2 Explanation

Explanation

c: A is invertible and A≠A^2 Explanation

Explanation

d: A is not invertible and A=A^3 Explanation

Explanation