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Solution of system of linear equations using linear map equation

Let

A=\begin{bmatrix} 2 & 3 \\ 0 & 0 \end{bmatrix}

. Then for which B the matrix equation AX=B is consistent

B=\begin{bmatrix} 0 \\ 3 \end{bmatrix}

Explanation

B=\begin{bmatrix} 0 \\ 1 \end{bmatrix}

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B=\begin{bmatrix} 1 \\ 0 \end{bmatrix}

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d: None of the above Explanation

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Let

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

. Then for which B the matrix equation AX=B is inconsistent

B=\begin{bmatrix} 0 \\ 3 \end{bmatrix}

Explanation

B=\begin{bmatrix} 0 \\ 0 \end{bmatrix}

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B=\begin{bmatrix} 4 \\ 4 \end{bmatrix}

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d: None of the above Explanation

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Consider the consistent matrix equation AX=B, where A is a 2×3 and B is a column matrix of order 2×1. Let T:R^3→R^2 be the linear transformation w.r.t. the standard basis associated with the matrix A. Then

a: B∈Range(T) Explanation

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b: B∈Null space(T) Explanation

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c: B=0 0 Explanation

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d: None of the above Explanation

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Consider the consistent matrix equation AX=B, where A is a 2×3 and B= 0 0. Let T:R^3→R^2 be the linear transformation w.r.t. the standard basis associated with the matrix A. Then

x \notin Range(T)

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x \notin Null space(T)

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c: X = (0, 0, 0) Explanation

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d: None of the above Explanation

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Consider a system of linear equations represented by AX=B, where A is an invertible matrix of order n×n. Then the system has

a: One and only one solution Explanation

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b: Infinitely many solutions Explanation

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c: Exactly two solutions Explanation

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d: Finite but more than two solutions Explanation

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Consider the system of two linear equations in three variables given by

x_1+x_2=3

and

x_1+x_2+x_3=0

. Then

a: System is consistent and has unique solution Explanation

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b: System is consistent and has finitely many solutions Explanation

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c: System is consistent and has infinitely many solutions Explanation

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d: System is inconsistent Explanation

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Consider the system of three linear equations in two variables given by

x_1+x_2=3, x_{1}-2x_2=-6

and

x_1+3x_2=6

. Then

a: System is consistent and has unique solution Explanation

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b: System is consistent and has finitely many solutions Explanation

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c: System is consistent and has infinitely many solutions Explanation

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d: System is inconsistent Explanation

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