- How do you multiply two matrices?
- Each entry of A × B is a row of A times a column of B: multiply the pairs and add them up. For A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], the top-left entry is 1·5 + 2·7 = 19, and the full answer is [[19, 22], [43, 50]]. That only works when A has as many columns as B has rows, and the result takes A's row count and B's column count, so a 2×3 times a 3×4 is a 2×4. Order matters too. B × A for the same pair is [[23, 34], [31, 46]], a different matrix.
- How is the determinant calculated?
- The calculator row-reduces A to an upper triangular matrix, multiplies the diagonal, and flips the sign once for every row swap. Adding a multiple of one row to another doesn't change the determinant, which is why this works. For [[2, −3, 1], [2, 0, −1], [1, 4, 5]], the steps are R2 → R2 − R1, R3 → R3 − 1/2·R1 and R3 → R3 − 11/6·R2. That leaves 2, 3 and 49/6 on the diagonal, and 2 · 3 · 49/6 = 49. On a big matrix this takes far fewer steps than cofactor expansion, and Show row operations lists every one.
- How do you find the inverse of a matrix?
- Write A next to the identity matrix as [A | I] and row-reduce until the left side becomes I. The right side is then A⁻¹. For [[1, 2], [3, 4]], that takes three steps: R2 → R2 − 3·R1, R2 → −1/2·R2, and R1 → R1 − 2·R2. The answer is [[−2, 1], [3/2, −1/2]]. For a 2×2 there's also a shortcut: swap the diagonal, negate the other two entries, and divide by the determinant, which is −2 here. Multiply A by the answer to check it, and you get the identity matrix.
- Why does it say my matrix has no inverse?
- Its determinant is 0, so it's singular. That happens when one row is a combination of the others, which squashes space into a lower dimension that can't be undone. In [[1, 2], [2, 4]], the second row is exactly twice the first. The first row operation, R2 → R2 − 2·R1, turns row 2 into zeros, so there's no pivot for column 2. The row operations still show, so you can see where the elimination got stuck. A matrix that isn't square never has an inverse, so the calculator asks you to match rows and columns first.
- What is RREF, and what does the rank tell me?
- Reduced row echelon form is where Gauss-Jordan elimination ends up: each pivot is a 1, it's the only non-zero entry in its column, and any rows of zeros sit at the bottom. For [[1, 2, 3], [4, 5, 6], [7, 8, 9]], the RREF is [[1, 0, −1], [0, 1, 2], [0, 0, 0]]. The rank is the number of pivots, 2 here, and it counts how many rows are truly independent. If a matrix is the coefficients of a system of equations, a rank lower than the number of unknowns means there are free variables: infinitely many solutions, or none.
- Why does it show fractions instead of decimals?
- Because fractions are exact, and linear algebra homework usually wants the exact answer. Every entry is stored as a fraction of two whole numbers, so nothing gets rounded along the way. Typing 0.1 and 0.2 into A + B gives exactly 3/10, not the 0.30000000000000004 that floating-point math produces. The inverse of [[2, 1], [1, 3]] comes out as [[3/5, −1/5], [−1/5, 2/5]]. Switch to Decimals and you see 0.6, −0.2, −0.2 and 0.4, rounded to at most 6 places. The switch only changes how the answer looks, never the math.
- Can I paste a matrix from Excel or Google Sheets?
- Yes. Copy a range, click any cell of A or B, and paste. The whole matrix is replaced and resized to match. Tabs, spaces and commas separate columns, while new lines and semicolons separate rows, so MATLAB-style [1 2; 3 4] works too. Short rows are padded with 0, and anything past 6 rows or 6 columns is dropped with a note. Going the other way, Copy writes tab-separated rows that paste straight into a spreadsheet, and Copy LaTeX writes a bmatrix like \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}. In a single cell, use a dot for decimals, since a comma is ambiguous.
- How do matrix powers work?
- Aⁿ is A multiplied by itself n times, so A³ = A × A × A. Only square matrices have powers, because A × A needs A's columns to match its rows. A⁰ is the identity matrix, and the calculator takes n from 0 to 20. Powers show up in Markov chains, graph walks and recurrences. The classic example is the Fibonacci matrix: [[1, 1], [1, 0]] to the 10th power is [[89, 55], [55, 34]], which holds the 11th, 10th and 9th Fibonacci numbers. For a negative power, find A⁻¹ first, press Use as A, then raise it to the positive power.
- What does the transpose do?
- It turns rows into columns. Row 1 of A becomes column 1 of Aᵀ, so a 2×3 matrix becomes a 3×2. For [[1, 2, 3], [4, 5, 6]], the transpose is [[1, 4], [2, 5], [3, 6]]. A matrix that equals its own transpose is called symmetric, like covariance matrices in statistics. One rule is worth knowing for proofs: the transpose of a product reverses the order, so (A × B)ᵀ = Bᵀ × Aᵀ. A matrix and its transpose always have the same determinant and the same rank.
