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If the matrix $\begin{pmatrix}1 & 2 & \sqrt{6}\0 & 0 & 1\0 & 0 & 0\end{pmatrix} \begin{pmatrix}3 & 0 & 1\6 & -1 & 2\2 & 0 & 1\end{pmatrix}$ is in reduced row echelon form (RREF), then √ is equal to:
If the matrix $\begin{pmatrix}1 & 2 & \sqrt{6}\0 & 0 & 1\0 & 0 & 0\end{pmatrix} \begin{pmatrix}3 & 0 & 1\6 & -1 & 2\2 & 0 & 1\end{pmatrix}$ is in reduced row echelon form (RREF), then √ is equal to:
- None
- -√6
- √6 (correct)
- 1+√6
- 1-√6
Consider the linear system: x + y + z = a, 2x + 3y + 3z = 16, 5x + 2y + 3z = 11. If x = -1, y = 2, and z = r is the solution of this system, where a ∈ ℝ and r ∈ ℝ, then a is equal to:
Consider the linear system: x + y + z = a, 2x + 3y + 3z = 16, 5x + 2y + 3z = 11. If x = -1, y = 2, and z = r is the solution of this system, where a ∈ ℝ and r ∈ ℝ, then a is equal to:
- -5
- 5
- None
- -4
- 4 (correct)
The linear system: x + 2y + 5z = 2, 4x + y - 15z = 15, 2x + 3y + 5z = 5 has:
The linear system: x + 2y + 5z = 2, 4x + y - 15z = 15, 2x + 3y + 5z = 5 has:
- no solution (correct)
- None
- a unique solution
- infinitely many solutions
- the trivial solution
If z = t, t ∈ ℝ, in the linear system: x + y + 3z = 2, 4x + 5y + 2z = 4, 3x + y + 29z = 14, y is equal to:
If z = t, t ∈ ℝ, in the linear system: x + y + 3z = 2, 4x + 5y + 2z = 4, 3x + y + 29z = 14, y is equal to:
In the linear system with z = t, t ∈ ℝ as in the previous question, x is equal to:
In the linear system with z = t, t ∈ ℝ as in the previous question, x is equal to:
In the linear system: x + 2y - z + 3w = -5, 2x + 5y + 2z + 7w = -3, 3x + 8y + 6z + 9w = -2, -2x - 5y + z -12w = -1, the value of w is:
In the linear system: x + 2y - z + 3w = -5, 2x + 5y + 2z + 7w = -3, 3x + 8y + 6z + 9w = -2, -2x - 5y + z -12w = -1, the value of w is:
The value of z is: _______________
The value of z is: _______________
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Study Notes
Matrix Operations
- A matrix is in reduced row echelon form (RREF) if it meets certain conditions.
- The matrix product [\begin{pmatrix} 1 & 2 & \sqrt{6}\ 0 & 0 & 1\ 0 & 0 & 0 \end{pmatrix} \begin{pmatrix} 3 & 0 & 1\ 6 & -1 & 2\ 2 & 0 & 1 \end{pmatrix}] is in RREF, and √ is equal to 1 + √6.
Linear Systems
- A linear system consists of multiple equations with variables.
- The linear system x + y + z = a, 2x + 3y + 3z = 16, 5x + 2y + 3z = 11 has a solution x = -1, y = 2, and z = r, where a ∈ ℝ and r ∈ ℝ, and a is equal to -5.
- The linear system x + 2y + 5z = 2, 4x + y - 15z = 15, 2x + 3y + 5z = 5 has infinitely many solutions.
- The linear system x + y + 3z = 2, 4x + 5y + 2z = 4, 3x + y + 29z = 14 has a solution z = t, t ∈ ℝ, and y is equal to -4 - 10t, and x is equal to 13 - 6t.
Linear System Applications
- The linear system x + 2y - z + 3w = -5, 2x + 5y + 2z + 7w = -3, 3x + 8y + 6z + 9w = -2, -2x - 5y + z -12w = -1 has a solution with a value of w equal to -1, and a value of z equal to 3.
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