JAMB Mathematics 2024 Paper

Given the matrix

$P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix},$

we want to find $P^2 - 4P - I$, where $I$ is the identity matrix:

$I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}.$

Calculate {$P^2$}

$P^2 = P \times P = \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix}.$

Calculate {$4P$}

$4P = 4 \times \begin{bmatrix} 1 & 2 \\ 2 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix}$

Calculate {$P^2 - 4P$}

$P^2 - 4P = \begin{bmatrix} 5 & 8 \\ 8 & 13 \end{bmatrix} - \begin{bmatrix} 4 & 8 \\ 8 & 12 \end{bmatrix} = \begin{bmatrix} 5 - 4 & 8 - 8 \\ 8 - 8 & 13 - 12 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$

Calculate {$P^2 - 4P - I$}

$P^2 - 4P - I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$

Thus, the result of $P^2 - 4P - I$ is

$\begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$