8.3 Determinants practice 1

Evaluate:

(a) [math]\begin{vmatrix}1&0\\3&7\end{vmatrix}[/math]

(b) [math]\begin{vmatrix}2&-9\\3&8\end{vmatrix}[/math]

(c) [math]\begin{vmatrix}1&-2&2\\3&-13&4\\9&0&9\end{vmatrix}[/math]

(d) [math]\begin{vmatrix}9&98&99\\0&8&97\\0&0&7\end{vmatrix}[/math]

Consider a matrix [math]A=\begin{bmatrix}a&-b\\c&-a\end{bmatrix}[/math] where [math]a, b[/math], and [math]c[/math] are real numbers. If [math]\det(A)=2[/math], what is the value of [math]\det(A+A^{-1})[/math]?

Evaluate:

(a) [math]\begin{vmatrix}-5&4\\-3&8\end{vmatrix}[/math]

(b) [math]\begin{vmatrix}1&0&2\\3&5&9\\-2&-3&0\end{vmatrix}[/math]

(c) [math]\begin{vmatrix}-4&3&-2\\3&0&3\\4&7&1\end{vmatrix}[/math]

Find [math]x[/math] such that [math]\begin{bmatrix}1-x&1\\6&2-x\end{bmatrix}[/math] is a singular matrix. If you have two or more answers, write the smallest one.

(Calculator) Let [math]A=\begin{bmatrix}1&-9&7&-1\\0&8&-4&-7\\1&4&5&0\\-3&-9&2&6\end{bmatrix}[/math]. Evaluate [math]C_{21}[/math].

Let [math]A(2,-1), B(4,0), C(-3,5)[/math] be three points on a plane. Find the area of the triangle [math]ABC[/math].

What is the product of [math]C_{32}[/math] and [math]M_{14}[/math], where [math]C_{ij}[/math] denotes the [math]i^{th}[/math] row [math]j^{th}[/math] column cofactor and [math]M_{ij}[/math] denotes the [math]i^{th}[/math] row [math]j^{th}[/math] column minor of matrix [math]A=\begin{bmatrix}1 & -1 & 3 & 4\\0 & 4 & -1 &0\\-1 & 2 & 1& 5\\-1& 4 & 6 & 1\end{bmatrix}[/math]

Given that [math]A=\begin{bmatrix}3 & -2\\-3 & 4 \end{bmatrix}[/math] and [math]I=\begin{bmatrix}1 & 0\\0 & 1 \end{bmatrix}[/math], find the maximum values of [math]\lambda[/math] for which [math](A-\lambda I)[/math] is a singular matrix. A matrix is singular if and only if its determinant value is zero.

The matrix [math]A[/math], [math]B[/math], [math]C[/math], and [math]X[/math] are all non-singular [math]3\times3[/math] matrices. Given that [math]A^{-1}XB=C[/math], express [math]X[/math] in terms of the other matrices.