CAIE A-Level Further Math 1.4.2 Determinants & inverses

Practise using determinants and inverses to test singularity, find parameter values and solve matrix equations in Paper 1 contexts.

Exam points

set det A=0 to find parameter values that make a square matrix singular
calculate inverses of non-singular 2 by 2 or 3 by 3 matrices and use them on the left
show a product matrix is singular by forming its determinant from the calculated entries

[Maximum number: 2]

Let $\mathbf{A}=\left(\begin{array}{ll}3 & 0 \\ 1 & 1\end{array}\right)$ .

Find, in terms of n, the inverse of $\mathbf{A}^{n}$ .

$\operatorname{det} \mathbf{A}^{n}=\operatorname{det}\left(\begin{array}{cc}3^{n} & 0 \frac{1}{2}\left(3^{n}-1\right) & 1\end{array}\right)=3^{n}$

Or a multiple of $\mathbf{A}^{-n}=2^{-1} 3^{-n}\left(\begin{array}{cc}2 & 0 1-3^{n} & 2 \times 3^{n}\end{array}\right)$ seen.

B1

$\mathbf{A}^{-n}=3^{-n}\left(\begin{array}{cc}1 \frac{1}{2}\left(1-3^{n}\right) 3^{n}\end{array}\right)$

B1

OE $\mathbf{A}^{-n}=2^{-1} 3^{-n}\left(\begin{array}{cc}2 1-3^{n} 2 \times 3^{n}\end{array}\right)$

2

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1.4.2—Determinants and inverses question 1