Our AI Matrices Solver specializes in the following matrix operations:

1. Matrix Notation: Understand the basics of matrix terminology and representation.
2. Matrix Addition and Subtraction: Perform addition and subtraction for 2×2 and 3×3 matrices.
3. Matrix Multiplication: Multiply matrices with step-by-step guidance for 2×2 and 3×3 matrices.
4. Unit (Identity) Matrix Recognition: Identify and understand the properties of identity matrices.
5. Determinants:
   • Calculate the determinant of a 2×2 matrix.
   • Calculate the determinant of a 3×3 matrix.
6. Inverses:
   • Determine the inverse (or reciprocal) of a 2×2 matrix.
   • Determine the inverse (or reciprocal) of a 3×3 matrix.

How It Works

Using the AI Matrices Solver is simple:

1. Select Your Operation:
   Choose from a list of matrix operations like addition, multiplication, determinant calculation, or inverse determination.

2. Enter Your Matrix or Matrices:
   Provide the required matrices (e.g., a single matrix for determinants or two matrices for addition/multiplication).

3. Get Step-by-Step Solutions:
   Watch as the solver breaks down each step, providing clear explanations and accurate results.

4. Learn and Apply:
   Understand the process behind the solution so you can apply the concepts to future problems.

1. Adding Two 2×2 Matrices

Problem: Add the matrices:
\[
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\quad \text{and} \quad
\begin{bmatrix}
5 & 6 \\
7 & 8
\end{bmatrix}.
\]

Solution:
– Add corresponding elements:
– First row: \(1 + 5 = 6\), \(2 + 6 = 8\).
– Second row: \(3 + 7 = 10\), \(4 + 8 = 12\).

Answer:
\[
\begin{bmatrix}
6 & 8 \\
10 & 12
\end{bmatrix}.
\]

2. Determinant of a 2×2 Matrix

Problem: Find the determinant of the matrix:
\[
\begin{bmatrix}
2 & 3 \\
4 & 5
\end{bmatrix}.
\]

Solution:
– Use the formula: \( \text{det} = (a \times d) – (b \times c) \).
– Compute: \( \text{det} = (2 \times 5) – (3 \times 4) = 10 – 12 = -2 \).

Answer:
\(-2\).

3. Multiplying Two 3×3 Matrices

Problem: Multiply the matrices:
\[
\begin{bmatrix}
1 & 0 & 2 \\
3 & 4 & 5 \\
6 & 7 & 8
\end{bmatrix}
\quad \text{and} \quad
\begin{bmatrix}
2 & 1 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}.
\]

Solution:
1. Multiply rows of the first matrix by columns of the second matrix:
– First row, first column: \(1 \times 2 + 0 \times 4 + 2 \times 7 = 16\).
– First row, second column: \(1 \times 1 + 0 \times 5 + 2 \times 8 = 17\).
– First row, third column: \(1 \times 3 + 0 \times 6 + 2 \times 9 = 21\).
– Repeat for all rows and columns.

Answer:
\[
\begin{bmatrix}
16 & 17 & 21 \\
61 & 64 & 75 \\
110 & 116 & 138
\end{bmatrix}.
\]

4. Inverse of a 2×2 Matrix

Problem: Find the inverse of the matrix:
\[
\begin{bmatrix}
4 & 7 \\
2 & 6
\end{bmatrix}.
\]

Solution:
1. Compute the determinant:
\( \text{det} = (4 \times 6) – (7 \times 2) = 24 – 14 = 10 \).
2. Use the inverse formula:
\[
\text{Inverse} = \frac{1}{\text{det}} \times
\begin{bmatrix}
d & -b \\
-c & a
\end{bmatrix}.
\]
3. Substitute values:
\[
\text{Inverse} = \frac{1}{10} \times
\begin{bmatrix}
6 & -7 \\
-2 & 4
\end{bmatrix}.
\]
4. Simplify:
\[
\text{Inverse} =
\begin{bmatrix}
0.6 & -0.7 \\
-0.2 & 0.4
\end{bmatrix}.
\]

Answer:
\[
\begin{bmatrix}
0.6 & -0.7 \\
-0.2 & 0.4
\end{bmatrix}.
\]

5. Determinant of a 3×3 Matrix

Problem: Find the determinant of the matrix:
\[
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6 \\
7 & 8 & 9
\end{bmatrix}.
\]

Solution:
1. Use the determinant formula:
\[
\text{det} = a(ei – fh) – b(di – fg) + c(dh – eg),
\]
where \(a, b, c\) are elements of the first row, and the rest correspond to the submatrices.
2. Compute:
\[
\text{det} = 1(5 \cdot 9 – 6 \cdot 8) – 2(4 \cdot 9 – 6 \cdot 7) + 3(4 \cdot 8 – 5 \cdot 7).
\]
– Compute sub-expressions:
\(5 \cdot 9 – 6 \cdot 8 = 45 – 48 = -3\),
\(4 \cdot 9 – 6 \cdot 7 = 36 – 42 = -6\),
\(4 \cdot 8 – 5 \cdot 7 = 32 – 35 = -3\).
– Substitute:
\[
\text{det} = 1(-3) – 2(-6) + 3(-3).
\]
– Simplify:
\[
\text{det} = -3 + 12 – 9 = 0.
\]

Answer:
\[
\text{det} = 0.
\]