Inverse matrices and systems of equations – further maths revision notes to download as PDF

Starter activity

Two phone plans are advertised. Plan A costs £10 a month plus 5p per minute, and plan B costs £4 a month plus 8p per minute. Write two equations for the monthly cost \(C\) in terms of the number of minutes \(m\), and solve them to find when the plans cost the same. Now imagine that there are three plans and three unknown quantities. Would you still be happy solving by substitution? Think about how a matrix could store the numbers in the equations, and about what operation on the matrix would give you the answer in a single step.

Note: when does an inverse exist?

A square matrix has an inverse exactly when its determinant is not zero. For a \(2\times2\) matrix this means that the two columns are not parallel, so the transformation does not squash the plane onto a line. For a \(3\times3\) matrix it means that the three columns do not lie in a common plane, so the transformation does not flatten space. When the determinant is zero the matrix is called singular, and information has been lost: many different points are sent to the same image, so there is no way of recovering the original. This is the reason why a singular matrix has no inverse, and also why a system of equations with a singular coefficient matrix either has no solutions or infinitely many.

Applications of matrix inverses

Whenever several unknown quantities are linked by several linear relationships, a matrix inverse provides the answer. Engineers use them to find the forces in the members of a bridge or the currents in a network. Economists use them in input-output models, where each industry buys from the others. Computer graphics programmes use inverse matrices to undo a transformation, for example when the mouse pointer is converted from screen coordinates back into the position in a three-dimensional scene. Cryptographers use them to decode messages that were scrambled by a matrix. In each case the structure is the same: a known linear transformation has been applied to an unknown input, and the inverse recovers the input from the output.

Why the order of multiplication matters

Matrix multiplication is not commutative, which means that \(\mathbf{AB}\) and \(\mathbf{BA}\) are usually different. This is the reason why you must be careful about the side on which you multiply when solving a matrix equation. If you have the equation \(\mathbf{AX}=\mathbf C\) and you want to isolate \(\mathbf X\), you need the product \(\mathbf A^{-1}\mathbf A\) to appear next to \(\mathbf X\), so you multiply both sides on the left by \(\mathbf A^{-1}\). If you multiplied on the right you would obtain \(\mathbf{AXA}^{-1}\), which does not simplify. In a similar way, for \(\mathbf{XA}=\mathbf C\) the inverse goes on the right. A good habit is to write the step explicitly, such as "pre-multiply both sides by \(\mathbf A^{-1}\)", so that the examiner can see that you know which side is correct. The same care is needed when expanding products: \((\mathbf A+\mathbf B)^2\) is \(\mathbf A^2+\mathbf{AB}+\mathbf{BA}+\mathbf B^2\), and you may not combine the two middle terms.

Reading the geometry from the algebra

The way a system behaves under elimination tells you the geometry directly. If you reach a line such as \(0=0\), one of the equations was redundant, and the solution has a free parameter, so the planes share a line or a whole plane. If you reach a false statement such as \(0=5\), the equations are inconsistent, so the planes have no common point. If you reach three independent equations for three unknowns, the planes meet at a single point. It is a good idea to write a short sentence in your answer that links the algebra to the geometry, because the question often asks for this explicitly, and an answer that merely states the number of solutions may not earn the final mark.

Using technology wisely

Graphical calculators and computer algebra systems can find inverses and solve systems in an instant, and in the examination you may use a calculator to check or to carry out a long calculation. However, a question that says "show that" or "find, by algebra" expects to see the working. Use your calculator to check the answer that you have found by hand, and to find the inverse of a three by three matrix when the question allows it, but write down the matrix you entered and the matrix that you obtained. Examiners cannot award marks for a number that appears from nowhere, and a mistake in typing an entry is easy to make.

Looking ahead

The ideas of this chapter lead directly to eigenvalues and eigenvectors, where you look for the special directions that a matrix leaves unchanged. They also lead to the study of vector spaces and linear algebra at university, to numerical methods for solving huge systems with thousands of equations, and to the matrix methods used in statistics and machine learning. A thorough grasp of inverses, determinants and the meaning of a singular matrix will be a great help in all of those areas.

Did you know?

The idea of a determinant appeared in China more than two thousand years ago, in a book called The Nine Chapters on the Mathematical Art, which solved systems of linear equations by a method very similar to the one we now call Gaussian elimination. In Europe, Gottfried Leibniz and Seki Takakazu of Japan found determinants independently in the 1680s. The word matrix, from the Latin for womb, was introduced by James Joseph Sylvester in 1850, and his friend Arthur Cayley developed the algebra of matrices, including the inverse, in a famous paper of 1858.