TL;DR: Type a matrix in the order the page names, usually row by row, left to right. Matrix multiplication is not commutative, so swapping two matrices is a different problem. A 3×3 page will not read a list of nine numbers correctly if you entered columns as rows.
Rows first, unless the page says otherwise
Mathematical Calculator’s linear-algebra pages cover operations such as addition, multiplication, determinants, inverses, rank, Gaussian elimination, and eigenvalues for small sizes like 2×2 and 3×3. Every one of those pages must decide how a grid becomes a list of boxes. The usual convention is row-major: finish row one before you start row two. If you instead type down the columns, you have entered the transpose. Determinants happen to agree for a matrix and its transpose. Products, inverses, and eigenvectors do not. A “small” mix-up in entry order is a different matrix.
Before the real data, type a matrix whose shape you can see, such as the 2×2 matrix with first row 1, 2 and second row 3, 4. If the page echoes the matrix in a grid, confirm that 2 sits to the right of 1 and not beneath it. If the page does not echo a grid, compute a determinant you know: for that example, 1×4 − 2×3 = −2. A different determinant means the boxes are not in the order you thought. Fix the order on this example. Do not “adjust” the real matrix until the example looks right.
Size is part of the input
A 2×2 addition page and a 3×3 addition page are different tools. Nine numbers forced into a four-box page will be truncated or rejected. Four numbers padded with a zero to fill a 3×3 page invent a different linear map. Read the size in the title and count your boxes before you calculate. For multiplication, the inner dimensions must agree: a 2×3 times a 3×2 is legal, and a 2×3 times a 2×3 is not. If the page only multiplies square matrices of one size, do not use it for a rectangular product and then wonder why the button does nothing.
Write the dimensions above the grid in your notes in the same order as the page. “Rows by columns” is the phrase that prevents you from counting the other way because a textbook drew the matrix as a tall table. Count the boxes on the screen. If you have three rows of three, you have a 3×3, even if you still think of it as “a nine-vector.”
Multiplication order is the question
AB and BA are different whenever both products exist, except in special cases you should not assume. A page labeled “A times B” must be given A and B in that order. Students who “put the second matrix first because the form listed B on top” compute the other product and get a clean, wrong matrix. Label the form’s slots with the letters from your notes before you type any entry. If the form has no labels, type one distinctive number, such as a 7 in the corner of A only, and confirm it lands in A’s corner on the echo.
The identity is the other handshake. A times the identity should return A. If it returns the transpose of A, your entry order and the page’s order disagree, and every later product will be quietly transposed. That single check is worth more than retyping the entries quickly and hoping.
Determinants, inverses, and what they do not forgive
- A row of zeros means the determinant is zero and no inverse exists. If the page returns an inverse anyway, a zero was typed into the wrong box.
- Swapping two rows flips the sign of the determinant. If your hand calculation disagrees by a sign, check whether you swapped rows on the way in.
- Gaussian elimination pages may show steps. Compare the first elimination step with one you do by hand. If the first pivot is wrong, the rest of the steps are a different system.
- Eigenvalue pages return values that depend on the actual matrix. Transposing does not change the eigenvalues, so this is a poor test of entry order. Use the identity product instead.
Cramer’s rule pages and inverse pages fail together when a system is singular. “No solution” or a zero determinant is an answer. It means the lines or planes do not meet in one point. Do not jiggle an entry by a rounding crumb to force an inverse. You would be solving a different system that the problem did not ask.
Fractions and decimals in the grid
If the page accepts fractions, prefer them for textbook matrices. Decimals invite early rounding, and an inverse will magnify that rounding. If the page wants decimals, convert fractions yourself and keep more digits than the final answer requires. A determinant that should be zero and comes back as 0.0001 is a rounding artifact or a mistyped entry. Substitute back into the original system before you declare the solution exact.
Negative signs need the same care as on any small keyboard. A minus that lands in the box to the right of the one you intended is a different matrix with a plausible determinant. Read the echoed grid, cell by cell, against the paper. This is slow once and faster than debugging a 3×3 product.
Copy the matrix back out
When you record the result, copy the output grid in row-major order and state the size. A list of nine numbers without row breaks cannot be checked. Circle the order you used if you are handing the work to someone who might read down the columns.
Choose the matrix tool from the Mathematical Calculator homepage by size and operation, run the 1, 2 / 3, 4 handshake, and only then enter the matrix from the assignment. The page will multiply whatever grid you gave it. Making that grid the one you meant is the part it cannot do for you.