(a)Calculatethedeterminantsofthefollowingmatrices, ifpossible(1p): A=   −2 1 3 1 0 3 0 1 3   (i) B= 3 4 2 −5 −6 −3 (ii) C=     2 3 6 −3 0 −2 −5 9 0 0 0 −4 0 0 0 1    . (iii) (b)Whichofthematricesfrompart(a)haveaninversematrix? (c)Forwhichsystemsofequationscorrespondingtothematricesinpart(a)isthesolution unique?

### Part (a): Determinants of the matrices

#### (i) Matrix A:
\[
A = \begin{bmatrix}
-2 & 1 & 3 \\
1  & 0 & 3 \\
0  & 1 & 3
\end{bmatrix}
\]

The determinant of a 3x3 matrix is calculated as:
\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]
Where the matrix is:
\[
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
\]

For matrix A, we substitute the values:
\[
a = -2, b = 1, c = 3, d = 1, e = 0, f = 3, g = 0, h = 1, i = 3
\]
Now we compute each part:

\[
ei - fh = (0)(3) - (3)(1) = 0 - 3 = -3
\]
\[
di - fg = (1)(3) - (3)(0) = 3 - 0 = 3
\]
\[
dh - eg = (1)(1) - (0)(0) = 1
\]

Now, substitute these values back into the determinant formula:

\[
\det(A) = -2(-3) - 1(3) + 3(1) = 6 - 3 + 3 = 6
\]
Thus, \(\det(A) = 6\).

#### (ii) Matrix B:
\[
B = \begin{bmatrix}
3 & 4 & 2 \\
-5 & -6 & -3
\end{bmatrix}
\]
This is a \(2 \times 3\) matrix, and determinants are only defined for square matrices. Therefore, \(\det(B)\) is **not defined**.

#### (iii) Matrix C:
\[
C = \begin{bmatrix}
2 & 3 & 6 & -3 \\
0 & -2 & -5 & 9 \\
0 & 0 & 0 & -4 \\
0 & 0 & 0 & 1
\end{bmatrix}
\]

The determinant of a \(4 \times 4\) matrix can be computed using cofactor expansion. Expanding along the first row (since the other rows have many zeros):
\[
\det(C) = 2 \cdot \det\begin{bmatrix}
-2 & -5 & 9 \\
0 & 0 & -4 \\
0 & 0 & 1
\end{bmatrix}
\]
The \(3 \times 3\) submatrix is upper triangular, so its determinant is the product of the diagonal elements:
\[
\det\begin{bmatrix}
-2 & -5 & 9 \\
0 & 0 & -4 \\
0 & 0 & 1
\end{bmatrix} = (-2) \cdot 0 \cdot 1 = 0
\]
Thus, \(\det(C) = 2 \times 0 = 0\).

### Part (b): Inverse of the matrices
A matrix has an inverse if and only if its determinant is non-zero.

- Matrix **A** has \(\det(A) = 6\), which is non-zero. Therefore, matrix A **has an inverse**.
- Matrix **B** is not square, so it **does not have an inverse**.
- Matrix **C** has \(\det(C) = 0\), so it **does not have an inverse**.

### Part (c): Systems with a unique solution
A system of equations corresponding to a matrix has a unique solution if the matrix is square and has a non-zero determinant.

- For matrix **A** (\(\det(A) = 6\)), the system has a **unique solution**.
- Matrix **B** is not square, so it does not represent a system with a unique solution.
- For matrix **C** (\(\det(C) = 0\)), the system does not have a unique solution; it may have no solutions or infinitely many solutions.

---

Do you have any questions or need further details on any part of the explanation? Here are 5 related questions you might find interesting:

1. How do you find the determinant of a 4x4 matrix using cofactor expansion?
2. What conditions must a matrix satisfy for it to have an inverse?
3. Why is the determinant important in solving systems of linear equations?
4. Can a non-square matrix ever have an inverse?
5. How does the rank of a matrix relate to the existence of a unique solution?

**Tip:** When working with larger matrices, simplifying calculations using properties like triangularity can save time.

Calculate the determinants of the following matrices, if possible: (i) A = [[-2, 1, 3], [1, 0, 3], [0, 1, 3]]; (ii) B = [[3, 4, 2], [-5, -6, -3]]; (iii) C = [[2, 3, 6, -3], [0, -2, -5, 9], [0, 0, 0, -4], [0, 0, 0, 1]]. Which of these matrices have an inverse matrix? For which systems of equations corresponding to the matrices is the solution unique?

Calculate the determinants of matrices A, B, and C, determine which matrices have an inverse, and identify systems with unique solutions. Explore matrix algebra concepts including determinants and inverse matrices for high school or college-level students.

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