# Problem-Solving: Linear Algebra

Physics I · Linear Algebra and Geometry · https://tryals.app/learn/physics-i/problem-solving-linear-algebra

## Linear Algebra Strategy

Mastering linear algebra requires connecting **algebraic** procedures with **geometric** intuition.

**4-Step Strategy**
1. **Identify** objects: vectors, matrices, transformations, or subspaces.
2. **Translate** into $A\vec{x}=\vec{b}$.
3. **Reduce** via Gaussian elimination or eigenvalues.
4. **Interpret** geometrically.

**Common Problem Types**

| Problem | Tool |
| --- | --- |
| Solve $A\vec{x}=\vec{b}$ | Row reduction |
| Is $A$ invertible? | $\det(A)\neq 0$ or rank $= n$ |
| Find eigenvalues | Characteristic polynomial |
| Diagonalise $A$ | $A = PDP^{-1}$ |
| Change basis | $[\vec{v}]_{B'} = P^{-1}[\vec{v}]_B$ |

## Checks and Pitfalls

**Key Insight**: Linear algebra problems reduce to one question: what does $A$ do to vectors? Eigenvalues and rank answer this completely.

**Pre-Submission Checklist**
- Verify dimensions match in every multiplication.
- Check: does $\det(A)=0$? If so, expect infinite or no solutions.
- Substitute your answer back into the original equation.

**Common Pitfall**: Invertibility is all-or-nothing. One collapsed direction (one zero eigenvalue, $\det = 0$, dependent columns) breaks every equivalent property at once. Check the cheapest condition and conclude the rest.

## Practice questions

7 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. The four statements below are **equivalent** for a square matrix $A$, each implies the next in a natural cycle of reasoning. Order the chain starting from the determinant.

**Answer:**

1. $\det A \ne 0$: the transformation preserves some volume
2. So no direction is collapsed: $Ax = 0$ only for $x = 0$
3. So the columns of $A$ are linearly independent
4. So every equation $Ax = b$ has exactly one solution

**Why:** This chain is a slice of the Invertible Matrix Theorem, a dozen conditions that live or die together. One number, $\det A$, answers all of them at once, which is why "check the determinant" is the first move on any square system.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/the-four-statements-below-are-equivalent-for-a-square-matrix-a-each

### 2. Rank, determinant, invertibility and independence keep turning out to be the same condition. Why does linear algebra have so many names for one thing?

A. Because each applies to a different size of matrix
B. Because each is the natural statement in a different setting
C. Because the terms were coined independently and the redundancy is historical accident
D. Because they agree only for square matrices and differ otherwise

**Answer:** B. Because each is the natural statement in a different setting

**Why:** The invertible matrix theorem collects the equivalences precisely because each phrasing is the useful one somewhere. Knowing they coincide is what makes the toolkit transferable.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/rank-determinant-invertibility-and-independence-keep-turning-out-to

### 3. If $A$ is a $3\times3$ matrix with rank 2, then the system $Ax = b$ has a unique solution for every $b$.

**Answer:** False

**Why:** Rank 2 < 3 means A is singular; the system may have no solution or infinitely many.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/if-a-is-a-3-3-matrix-with-rank-2-then-the-system-ax-b-has-a-unique

### 4. Complete the statement with the correct term.

**Answer:** The rank of a matrix equals the number of **pivots** in its row echelon form.

**Why:** Pivots are the leading nonzero entries; their count equals the rank.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/complete-the-statement-with-the-correct-term

### 5. Which are equivalent to saying A is invertible (for square A)?

A. $det(A)$ ≠ 0
B. rank(A) = n
C. The only solution of $Ax = 0$ is $x = 0$
D. A has a zero eigenvalue

**Answer:** A. $det(A)$ ≠ 0; B. rank(A) = n; C. The only solution of $Ax = 0$ is $x = 0$

**Why:** A zero eigenvalue means singular; the other three are equivalent to invertibility.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/which-are-equivalent-to-saying-a-is-invertible-for-square-a

### 6. Complete the Invertible Matrix Theorem summary.

**Answer:** A square matrix A is **invertible** if and only if det(A) is **nonzero** and A has **full** rank.

**Why:** Full rank, nonzero determinant, and invertibility are all equivalent.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/complete-the-invertible-matrix-theorem-summary

### 7. Sort these into properties of Invertible or Singular matrices.

**Answer:**

- Invertible: $det(A)$ ≠ 0, The rows span $\mathbb{R}^{n}$, $Ax = 0$ has only the trivial solution
- Singular: Has a zero eigenvalue, rank(A) < n

**Why:** The Invertible Matrix Theorem gives many equivalent conditions.

Page: https://tryals.app/practice/physics-i/problem-solving-linear-algebra/sort-these-into-properties-of-invertible-or-singular-matrices