3.2 Exercises

20 points · 4 extra credit

Properties of Determinants

1. 1. (3 pts) Compute $\det A$ by reducing $A$ to echelon form using only row operations, and state which operations you used and how each affects the determinant.

   $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 5 & 3 \\ 1 & 0 & 8 \end{bmatrix}.$

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2. 2. (3 pts) Reduce the matrix below to echelon form to evaluate its determinant. Your reduction must include at least one row interchange; track its effect on the value.

   $A = \begin{bmatrix} 1 & 3 & 1 & 1 \\ 2 & 5 & 2 & 2 \\ 1 & 3 & 8 & 9 \\ 1 & 3 & 2 & 2 \end{bmatrix}.$

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3. 3. (2 pts) Let $A$ be a $3\times 3$ matrix with $\det A = 4$. Find the determinant of each matrix obtained from $A$ by a single change:

   (a) multiply row 2 by $5$;

   (b) interchange rows 1 and 3;

   (c) form $2A$ (scale the entire matrix by $2$);

   (d) add $3$ times row 1 to row 2.

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4. 4. (3 pts) Suppose $A$ and $B$ are $n\times n$ matrices with $\det A = 3$ and $\det B = -2$. Compute:

   (a) $\det(AB)$; (b) $\det(A^{\mathsf T})$; (c) $\det(A^{-1})$; (d) $\det(A^2 B)$; (e) $\det(B^{-1}AB)$.

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5. 5. (2 pts) Let $A$ be a $5\times 5$ matrix with $\det A = 3$. Find $\det(2A)$ and $\det(-A)$.

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6. 6. (3 pts) State whether each statement is true or false, and justify each answer.

   (a) $\det(A + B) = \det A + \det B$ for all $n\times n$ matrices $A$ and $B$.

   (b) $\det(cA) = c^n\,\det A$ for every scalar $c$ and every $n\times n$ matrix $A$.

   (c) If $A$ is invertible, then $\det(A^{\mathsf T}) = \det(A^{-1})$.

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7. 7. (2 pts) Find all values of $k$ for which the matrix below is invertible.

   $A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & 2 & k \\ 1 & 4 & k^2 \end{bmatrix}.$

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8. 8. (2 pts) Let $A$ be an invertible $n\times n$ matrix. Using the multiplicative property $\det(AB) = (\det A)(\det B)$, show that $\det A \neq 0$ and that $\det(A^{-1}) = \dfrac{1}{\det A}$.

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9. 9. (4 pts ••) Prove that a square matrix $A$ is invertible if and only if $\det A \neq 0$. Use the fact that $A$ can be reduced to an echelon (upper-triangular) form $U$ using only row interchanges and row replacements, and relate $\det A$ to the pivots of $U$.

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