E600 Mathematics

MSc Economics Preparatory Module

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Chapter 2 – Quiz 2

1. Consider the matrix $A = \begin{pmatrix} 4 & 1\\ -1 & 3\end{pmatrix}$ . The matrix is…

negative semi-definite but not negative definite.

positive semi-definite but not positive definite.

positive definite.

indefinite.

negative definite.

2. True or false: any identity matrix is its own inverse.

True.

False.

3. Which statements about matrix inversion are true? (Multiple choice)

If $A,B$ are invertible matrices, then $(AB)^{-1} = A^{-1}B^{-1}$ .

Any diagonal matrix with non-zero trace is invertible.

The Gauss-Jordan algorithm uses only Elementary Matrix Operations to compute the inverse of an invertible matrix.

There are invertible matrices with a determinant equal to zero.

Any negative definite matrix is not invertible.

Any positive definite matrix is invertible.

4. Consider $A = \begin{pmatrix}1 & 0 & 1\\ 0 & 1 & 0\\-1 & 2 & -1\end{pmatrix}$ . What is the determinant of $A$ ? Is $A$ invertible?

-2; No.

0; No.

0; Yes.

-2; Yes.

5. Consider $A=\begin{pmatrix}2 & 0\\ 1 & -4\end{pmatrix}$ . Invert this matrix. What is the smallest eigenvalue of the inverse $A^{-1}$ ?

-4

-2

6. Consider two invertible matrices $A, B$ . How can you simplify $[(A\cdot B)']^{-1}$ ? (Hint: recall the rule for the transpose of a product given early in Chapter 2.)

$(B')^{-1}\cdot (A')^{-1}$

$(A')^{-1}\cdot (B')^{-1}$

$A\cdot (B')^{-1}$

$B\cdot (A')^{-1}$

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