Solution of \mathbf{A}\mathbf{x}=\mathbf{b} when \det\mathbf{A}\neq0?

\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}

What does \det\mathbf{A}\neq0 mean geometrically?

The three planes meet at a single point.

What does \det\mathbf{A}=0 tell you about the number of solutions?

There is not a unique solution: either infinitely many or none.

What is a consistent system?

One with at least one solution.

How can you tell if a singular system is consistent?

Eliminate; a true statement like 0=0 means consistent, a contradiction like 5=7 means inconsistent.

Three planes meeting in a common line?

A sheaf: infinitely many solutions.

Three planes with no common point, no two parallel?

A triangular prism: no solution.

Name two other arrangements with no solution.

Two parallel planes with a third crossing them; three distinct parallel planes.

What arrangements of coincident planes give infinitely many solutions?

Two coincident planes crossed by a third (a line), or all three coincident (a plane).

How do you give infinitely many solutions?

Let one variable be a parameter t and express the other two in terms of t.

Solving simultaneous equations with matricesAQA A-Level Further Maths: Flashcards

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How do you write three simultaneous equations as a matrix equation?

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How do you write three simultaneous equations as a matrix equation?: $\mathbf{A}\mathbf{x}=\mathbf{b}$ , with the coefficients as rows of $\mathbf{A}$ .
Solution of $\mathbf{A}\mathbf{x}=\mathbf{b}$ when $\det\mathbf{A}\neq0$ ?: $\mathbf{x}=\mathbf{A}^{-1}\mathbf{b}$
What does $\det\mathbf{A}\neq0$ mean geometrically?: The three planes meet at a single point.
What does $\det\mathbf{A}=0$ tell you about the number of solutions?: There is not a unique solution: either infinitely many or none.
What is a consistent system?: One with at least one solution.
What is an inconsistent system?: One with no solution.
How can you tell if a singular system is consistent?: Eliminate; a true statement like $0=0$ means consistent, a contradiction like $5=7$ means inconsistent.
Three planes meeting in a common line?: A sheaf: infinitely many solutions.
Three planes with no common point, no two parallel?: A triangular prism: no solution.
Name two other arrangements with no solution.: Two parallel planes with a third crossing them; three distinct parallel planes.
What arrangements of coincident planes give infinitely many solutions?: Two coincident planes crossed by a third (a line), or all three coincident (a plane).
How do you give infinitely many solutions?: Let one variable be a parameter $t$ and express the other two in terms of $t$ .

Exam questions on Solving simultaneous equations with matrices

Three planes have equations $x+y+z=6$ , $2x-y+z=6$ and $x+3y-z=2$ . A calculator may be used.
The inverse of the coefficient matrix is $\frac18\begin{pmatrix}-2&4&2\\3&-2&1\\7&-2&-3\end{pmatrix}$ . Use it to solve the equations.2 marks
Three planes have equations $x+y-z=2$ , $2x+2y-2z=4$ and $x-y+z=0$ .
Find the general solution of the three equations.2 marks
Three planes have equations $x+2y+z=4$ , $2x+y-z=1$ and $3x+3y=k$ , where $k$ is a constant.
Show that the three equations have no solution when $k=7$ , and state the geometrical arrangement of the planes.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).