Revise Matrices: determinants, inverses and applications for IB Mathematics: Applications and Interpretation (1.6)

A matrix is described by its order, rows by columns. You add or subtract two matrices only when they have the same order, combining entries in matching positions. Scalar multiplication multiplies every entry by the same number. Multiplication is the operation that surprises people: to multiply $A$ by $B$ you take rows of $A$ against columns of $B$, so the inner dimensions must match. The rule for this is called conformability.

Conformability: $(m \times n) \times (n \times p) \rightarrow (m \times p)$. \n The two INNER numbers must agree ($n = n$); the OUTER numbers give the order of the product.

Because order matters, keep the sequence of any product exactly as written. This single fact, that $AB \neq BA$, is the reason we must be careful when we later 'pre-multiply' both sides of a matrix equation by an inverse.

Among square matrices, one plays the role that 1 plays for numbers: the identity matrix $I$, with 1s on the leading diagonal and 0s everywhere else. Multiplying any conformable matrix by $I$ leaves it unchanged, $AI = IA = A$. The identity is what an inverse produces: $A^{-1}$ is precisely the matrix for which $AA^{-1} = A^{-1}A = I$, so the identity is the target every inverse aims at.

The $2 \times 2$ identity: $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$, with $AI = IA = A$. \n An inverse satisfies $AA^{-1} = A^{-1}A = I$.

Every square matrix carries a single number called its determinant, written $\det A$ or $|A|$, that decides whether the matrix can be inverted. For a $2 \times 2$ matrix you compute it by hand; for a $3 \times 3$ you use the GDC. Whatever the size, the meaning is the same: a non-zero determinant means an inverse exists, a zero determinant means it does not.

For a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$: \n $\det A = ad - bc$ (a scalar, not a matrix).

The inverse $A^{-1}$ undoes the effect of $A$: multiply them together, in either order, and you get the identity. For a $2 \times 2$ matrix there is a compact formula in the booklet; for a $3 \times 3$ you use the GDC. In both cases the inverse exists only when $\det A \neq 0$.

Inverse of a 2x2 matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$: \n $A^{-1} = \dfrac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad - bc \neq 0$.

The headline application of inverses is solving simultaneous linear equations. Line the variables up in the same order in each equation and the whole system collapses into one matrix equation $A\mathbf{x} = \mathbf{b}$, with $A$ the coefficients, $\mathbf{x}$ the unknowns and $\mathbf{b}$ the constants.

You cannot divide by a matrix, so instead pre-multiply both sides by $A^{-1}$. Because $A^{-1}A = I$, this isolates $\mathbf{x}$.

$A\mathbf{x} = \mathbf{b} \ \Rightarrow\ A^{-1}(A\mathbf{x}) = A^{-1}\mathbf{b} \ \Rightarrow\ (A^{-1}A)\mathbf{x} = A^{-1}\mathbf{b} \ \Rightarrow\ I\mathbf{x} = A^{-1}\mathbf{b} \ \Rightarrow\ \mathbf{x} = A^{-1}\mathbf{b}.$

Encoding a message uses the inverse as a decoding key. Letters are converted to numbers and arranged in a matrix, then multiplied by a non-singular key matrix to scramble them; the recipient multiplies the scrambled matrix by the inverse key to recover the original. The determinant must be non-zero, otherwise the scramble cannot be undone. Geometric transformations use matrices differently: multiplying a point's position vector by a suitable matrix maps it to a new point, so rotations, reflections, enlargements and shears each have a matrix, and composing transformations corresponds to multiplying their matrices (in reverse order of application). The determinant of a transformation matrix even carries meaning, its magnitude is the area scale factor of the transformation.

On Paper 1 and Paper 2 alike the marks are analytic: each is tied to a specific line of working, a method mark (M) or an accuracy mark (A), and an accuracy mark depends on the method mark it follows. Follow-through (FT) means a value that is wrong earlier does not have to cost you the marks that depend on it, provided the later step is done correctly on your own figure. The engine also ignores subsequent working (ISW) once a correct answer appears, and accepts any equivalent form and any correctly-rounded value, but only if the method is on the page. The GDC is expected for a $3 \times 3$; a $2 \times 2$ inverse should be shown by formula. Study how each mark below is earned by a specific line.

Matrices are the backbone of the rest of the HL number-and-algebra strand and beyond. The eigenvalues and eigenvectors you meet later are built directly on determinants and the equation $A\mathbf{x}=\lambda\mathbf{x}$; the transition matrices of Markov chains multiply exactly as here, with the long-run state read from a special vector; and systems of differential equations are solved by diagonalising a coefficient matrix. Even the geometric transformations touched on above return when you model repeated processes or graphics. Master the three moves of this lesson, check conformability, test the determinant, and solve $A\mathbf{x}=\mathbf{b}$ with the inverse in the correct order, and the more advanced matrix topics become variations on a method you already own.