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Which one of these has a Invertible matrix

To determine which of the given matrices is invertible, we must check if their determinants are non-zero. Only matrices with a non-zero determinant are invertible.

Let's go through each matrix:

1. **Matrix a)**

\[
A = \begin{pmatrix} 
1 & 3 \\ 
4 & 5 
\end{pmatrix}
\]

The determinant of a 2x2 matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by:

\[
\text{det}(A) = ad - bc
\]

For matrix a):

\[
\text{det}(A) = (1 \times 5) - (3 \times 4) = 5 - 12 = -7
\]

Since the determinant is non-zero, **Matrix a is invertible**.

---

2. **Matrix b)**

\[
A = \begin{pmatrix} 
-1 & 1 & 1 \\ 
1 & -1 & 1 \\ 
1 & 1 & -1 
\end{pmatrix}
\]

The determinant of a 3x3 matrix \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \) is given by:

\[
\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For matrix b):

\[
\text{det}(A) = (-1)((-1)(-1) - (1)(1)) - (1)((1)(-1) - (1)(1)) + (1)((1)(1) - (-1)(1))
\]

\[
= (-1)(1 - 1) - (1)(-1 - 1) + (1)(1 + 1)
\]
\[
= 0 + 2 + 2 = 4
\]

Since the determinant is non-zero, **Matrix b is invertible**.

---

3. **Matrix c)**

\[
A = \begin{pmatrix} 
1 & 2 & 3 \\ 
2 & 4 & 5 \\ 
3 & 5 & 6 
\end{pmatrix}
\]

Let's calculate the determinant:

\[
\text{det}(A) = 1((4)(6) - (5)(5)) - 2((2)(6) - (3)(5)) + 3((2)(5) - (3)(4))
\]
\[
= 1(24 - 25) - 2(12 - 15) + 3(10 - 12)
\]
\[
= 1(-1) - 2(-3) + 3(-2)
\]
\[
= -1 + 6 - 6 = -1
\]

Since the determinant is non-zero, **Matrix c is also invertible**.

---

### Conclusion:
All three matrices (a, b, and c) are invertible, as their determinants are non-zero.

Would you like details on how to compute the inverse for one of these matrices or have further questions?

Here are 5 related questions for deeper exploration:
1. How do you compute the inverse of a 2x2 matrix?
2. What is the general formula for finding the inverse of a 3x3 matrix?
3. Can a matrix with a zero determinant be invertible?
4. What are the properties of invertible matrices?
5. How can the inverse of a matrix be used in solving systems of linear equations?

**Tip:** To quickly check whether a matrix is invertible, compute its determinant. If it's zero, the matrix is singular and has no inverse.

Which of the matrices a), b), or c) has an invertible matrix?

Learn how to check whether a matrix is invertible by calculating its determinant. This page includes step-by-step calculations for three different matrices, covering both 2x2 and 3x3 matrices.

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