AI Matrix Solver

Core matrix operations

• Addition: add matching entries; the matrices must be the same size.
• Multiplication: each entry is a row of the first multiplied by a column of the second. The number of columns in the first must equal the number of rows in the second.
• Determinant: a single number summarising a square matrix. For [[a,b],[c,d]] it is ad − bc.
• Inverse: the matrix that multiplies to give the identity. It exists only when the determinant is not zero.

How to enter a matrix

Write rows inside double brackets, for example Find the determinant of [[2,1],[4,3]]. Each inner bracket is a row. Matrix questions are handled by the AI mode.

Matrices and equations

A system such as 2x + y = 5 and 4x + 3y = 11 can be written as AX = B and solved with X = A⁻¹B. This links directly to the system of equations solver. Determinants also give areas and volumes, which connects to geometry.

Worked examples

Example 1: Find the determinant of [[2,1],[4,3]]

1. Use ad − bc with a = 2, b = 1, c = 4, d = 3.
2. 2 × 3 − 1 × 4 = 6 − 4 = 2.

Final answer: 2

Example 2: Find the inverse of [[2,1],[4,3]]

1. The determinant is 2, so the inverse exists.
2. Swap a and d, negate b and c: [[3,−1],[−4,2]].
3. Divide by the determinant: ½ × that matrix.
4. Check the top row against the original: 2(3/2) + 1(−2) = 1 and 2(−1/2) + 1(1) = 0 ✓

Final answer: [[3/2, −1/2], [−2, 1]]

Example 3: Multiply [[1,2],[3,4]] by [[0,1],[1,0]]

1. Row 1 × columns: 1·0 + 2·1 = 2 and 1·1 + 2·0 = 1.
2. Row 2 × columns: 3·0 + 4·1 = 4 and 3·1 + 4·0 = 3.

Final answer: [[2, 1], [4, 3]]

Example 4: Find the determinant of [[1,2,3],[0,1,4],[5,6,0]]

1. Expand along the first row: 1(1·0 − 4·6) − 2(0·0 − 4·5) + 3(0·6 − 1·5).
2. = 1(−24) − 2(−20) + 3(−5) = −24 + 40 − 15.

Final answer: 1

Common mistakes to avoid

• Assuming matrix multiplication is commutative; AB is usually not BA.
• Multiplying entry by matching entry instead of row by column.
• Trying to invert a matrix whose determinant is zero.
• Forgetting the alternating plus and minus signs in a 3×3 determinant.
• Multiplying matrices whose sizes do not match.

Frequently asked questions

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