Collection of Maths Problems

Equations with singular matrices

Task number: 2430

• Variant 1

  \(\mathbf A= \begin{pmatrix} 1 &0 &1\\ 3 &0 &2\\ 2 &2 &0\\ 0 &2 &2 \end{pmatrix} \), \( \mathbf B= \begin{pmatrix} 3 &-1\\ 8 &-3\\ 10 & 0\\ 8 & 2 \end{pmatrix} \).

• Hint

  The elementary operations on the augmented matrix \((\mathbf A|\mathbf B)\) do not change the solution \(\mathbf X\).

• Resolution

  \( \begin{pmatrix} 1 &0 &1 &|& 3 &-1\\ 3 &0 &2 &|& 8 &-3\\ 2 &2 &0 &|&10 & 0\\ 0 &2 &2 &|& 8 & 2 \end{pmatrix} \sim…\sim \begin{pmatrix} 1 & 0 & 0 &|& 2 &-1\\ 0 & 1 & 0 &|& 3 & 1\\ 0 & 0 & 1 &|& 1 & 0\\ 0 & 0 & 0 &|& 0 & 0 \end{pmatrix} \)

  Hence \( \mathbf X= \begin{pmatrix} 2 &-1\\ 3 & 1\\ 1 & 0\\ \end{pmatrix} \).

• Variant 2

  \(\mathbf A= \begin{pmatrix} 1 &0 &1\\ 3 &0 &3\\ 2 &2 &0\\ 0 &2 &2 \end{pmatrix} \), \( \mathbf B= \begin{pmatrix} 3 &-1\\ 8 &-3\\ 10 & 0\\ 8 & 2 \end{pmatrix} \).

• Resolution

  \( \begin{pmatrix} 1 &0 &1 &|& 3 &-1\\ 3 &0 &3 &|& 8 &-3\\ 2 &2 &0 &|&10 & 0\\ 0 &2 &2 &|& 8 & 2 \end{pmatrix} \sim…\sim \begin{pmatrix} 1 & 0 & 0 &|& 0 &-1\\ 0 & 1 & 0 &|& 0 & 1\\ 0 & 0 & 1 &|& 0 & 0\\ 0 & 0 & 0 &|& 1 & 0 \end{pmatrix} \)

  Hence the equation has no solution.

• Variant 3

  \( \begin{pmatrix} 1 &0 &1\\ 3 &0 &3\\ 2 &-1 &0\\ 0 &1 &2 \end{pmatrix} \), \( \mathbf B= \begin{pmatrix} 3 &-1\\ 9 &-3\\ 1 &-3\\ 5 & 1 \end{pmatrix} \).

• Resolution

  \( \begin{pmatrix} 1 &0 &1 &|& 3 &-1\\ 3 &0 &3 &|& 9 &-3\\ 2 &-2&0 &|&-2 &-4\\ 0 &2 &2 &|& 8 & 2 \end{pmatrix} \sim…\sim \begin{pmatrix} 1 & 0 & 1 &|& 3 &-1\\ 0 & 1 & 2 &|& 5 & 1\\ 0 & 0 & 0 &|& 0 & 0\\ 0 & 0 & 0 &|& 0 & 0 \end{pmatrix} \)

  Hence \( \mathbf X= \begin{pmatrix} 3-p &-1-q\\ 5-2p & 1-2q\\ p & q\\ \end{pmatrix} \)