Which of the matrices a), b), or c) has an invertible matrix?
Solution
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:
Matrix a)
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.
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Math Problem Analysis
Mathematical Concepts
Linear Algebra Matrices Determinants Invertibility
Formulas
Determinant of a 2x2 matrix: det(A) = ad - bc Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
A matrix is invertible if its determinant is non-zero