HexMathv1.0.3

Matrix calculator

Type or scan a matrix problem. HexMath calculates it entry by entry and shows each step.

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What this page covers

  • Adding, subtracting and scaling matrices of the same size, and the transpose.
  • Determinants of 2×2 and 3×3 matrices.
  • Rank and reduced row echelon form by row operations.
  • Not covered here in detail: products (use Matrix multiplication calculator), inverses (use Inverse matrix calculator) and eigenvalues (use Eigenvalue calculator).

How to enter the problem

  • Type it or paste it. Write each matrix row by row, for example [[3, 1], [−2, 5]], or use the math keyboard's matrix key, and name the operation.
  • Scan it. Fit the one problem in the frame.
  • Choose a photo. Several problems in one photo are solved as a list.
  • Then tap Solve.

Worked examples

Example 1

Problem

Find the determinant of (2−1304112−2)\displaystyle \begin{pmatrix}2 & -1 & 3\\ 0 & 4 & 1\\ 1 & 2 & -2\end{pmatrix}.

Answer

Verified

The answer is

−33-33

Explanation

  1. Expand along the first row
    det⁡=2∣412−2∣−(−1)∣011−2∣+3∣0412∣\det = 2\begin{vmatrix}4 & 1\\ 2 & -2\end{vmatrix} - (-1)\begin{vmatrix}0 & 1\\ 1 & -2\end{vmatrix} + 3\begin{vmatrix}0 & 4\\ 1 & 2\end{vmatrix}
  2. Find the three 2×2 determinants
    4(−2)−1⋅2=−104(-2) - 1 \cdot 2 = -10, 0(−2)−1⋅1=−10(-2) - 1 \cdot 1 = -1, 0⋅2−4⋅1=−40 \cdot 2 - 4 \cdot 1 = -4.
  3. Combine
    2(−10)+1(−1)+3(−4)=−20−1−12=−332(-10) + 1(-1) + 3(-4) = -20 - 1 - 12 = -33

Example 2

Problem

Find the reduced row echelon form of (12−1251)\displaystyle \begin{pmatrix}1 & 2 & -1\\ 2 & 5 & 1\end{pmatrix}.

Answer

Verified

The answer is

(10−7013)\begin{pmatrix}1 & 0 & -7\\ 0 & 1 & 3\end{pmatrix}

Explanation

  1. Clear below the first pivot
    R2→R2−2R1R_2 \to R_2 - 2R_1 gives the row (0,1,3)(0, 1, 3).
  2. Clear above the second pivot
    R1→R1−2R2R_1 \to R_1 - 2R_2 gives the row (1,0,−7)(1, 0, -7).
  3. Read the result
    Each pivot is 1 and is the only nonzero entry in its column.

Example 3

Problem

Find the rank of (123246101)\displaystyle \begin{pmatrix}1 & 2 & 3\\ 2 & 4 & 6\\ 1 & 0 & 1\end{pmatrix}.

Answer

Verified

The answer is

rank⁡(A)=2\operatorname{rank}(A) = 2

Explanation

  1. Spot a dependent row
    Row 2 is twice row 1, so R2→R2−2R1R_2 \to R_2 - 2R_1 turns it into a row of zeros.
  2. Reduce the third row
    R3→R3−R1R_3 \to R_3 - R_1 gives (0,−2,−2)(0, -2, -2), which is not zero.
  3. Count the nonzero rows
    Two nonzero rows remain in echelon form.

Common mistakes

  • Subtracting only some entries, or subtracting in the wrong order. A−BA - B subtracts each entry of BB from the matching entry of AA.
  • Forgetting the alternating signs in a cofactor expansion. The middle term of a 3×3 expansion along the first row has a minus sign.
  • Adding matrices of different sizes. Addition and subtraction need the same number of rows and columns.
  • Stopping at row echelon form when the question asks for reduced row echelon form, where every pivot column is cleared above and below.

Checks, assumptions and limits

  • Check a determinant by expanding along a different row or column. The value must be the same.
  • Entries are kept as exact integers or fractions. Determinants, inverses and eigenvalues need square matrices.
  • HexMath can make mistakes. Double check important steps.

Related tasks

Frequently asked questions

Addition, subtraction, scaling, transpose, determinant, rank and reduced row echelon form. You can also ask for a product, inverse or eigenvalues and see the steps.

Write it row by row in square brackets, such as [[1, 2], [3, 4]], or use the math keyboard's matrix key. Then name the operation.

Yes. Use the camera on the one problem, or choose a photo. Several problems in one photo are solved as a list.

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