Suppose A is n × n and the equation Ax = 0 has only the trivial solution. Explai

Question

Suppose A is n × n and the equation Ax = 0 has only the trivial solution. Explain why A has n pivot columns and A is row equivalent to In Choose the correct answer below A on, thus A has n vo co umns. Since A s Suppose A is n × n and the equation Ax = 0 has only he rivial Solution Then there are n ree square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the nxn identity matrix, In arables n nse B. Suppose A is n × n and the equation Ax =0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in an echelon form of A must be on the main diagonal. Hence A is row equivalent to the nxn identity matrix, n C. Suppose A is n x n and the equation Ax = 0 has only the trivial solution. Then there are no free variables in this equation, thus A has n pivot columns. Since A is square and the n pivot positions must be in different rows, the pivots in A must be on the main diagonal. Hence A is the n xn identity matrix, I

Explanation / Answer

Ax = 0 has only trivial solution iff A is invertible i.e. A has n independent columns.

It is nx n matrix .

So it have n basic variables so it have no free variables.

So it have n pivot columns, we know pivots are determined by converting the matrix to row echelon form.

Since n is a square matrix so n pivot position must be in different rows and due to row echelon form pivots in A must be on the main diagonal only.

Hence, A is nxn identity matrix.

Thus option b is true

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