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" Technically, there is no such thing as matrix division. Dividing a matrix by another matrix is an undefined function. The closest equivalent is multiplying by the inverse of another matrix. In other words, while [A] [B] is undefined, you can solve the problem [A] * [B]-1. Since these two equations would be equivalent for scalar quantities, this "feels" like matrix division, but it's important to use the correct terminology. Note that [A] * [B]-1 and [B]-1 * [A] are not the same problem. You may need to solve both to find all possible solutions. For example, instead of (13263913)(7423){\displaystyle {egin{pmatrix}13&26\39&13\end{pmatrix}}\div {egin{pmatrix}7&4\2&3\end{pmatrix}}}, write (13263913)(7423)1{\displaystyle {egin{pmatrix}13&26\39&13\end{pmatrix}}*{egin{pmatrix}7&4\2&3\end{pmatrix}}^{-1}}.You may also need to calculate (7423)1(13263913){\displaystyle {egin{pmatrix}7&4\2&3\end{pmatrix}}^{-1}*{egin{pmatrix}13&26\39&13\end{pmatrix}}}, which may have a different answer. To take the inverse of a matrix, it must be a square matrix, with the same number of rows and columns. If the matrix you're planning to inverse is non-square, there is no unique solution to the problem. The term "divisor matrix" is a little loose, since this is not technically a division problem. For [A] * [B]-1, this refers to matrix [B]. In our example problem, this is (7423){\displaystyle {egin{pmatrix}7&4\2&3\end{pmatrix}}}. A matrix that has an inverse is called "invertible" or "non-singular." Matrices without an inverse are "singular." To multiply two matrices together, the number of columns in the first matrix must equal the number of rows in the second matrix. If this does not work in either arrangement ([A] * [B]-1 or [B]-1 * [A]), there is no solution to the problem. For example, if [A] is a 4 x 3 matrix (4 rows, 3 columns) and [B] is a 2 x 2 matrix (2 rows, 2 columns), there is no solution. [A] * [B]-1 does not work since 3 = 2, and [B]-1 * [A] does not work since 2 = 4. Note that the inverse [B]-1 always has the same number of rows and columns as the original matrix [B]. There's no need to calculate the inverse to complete this step. In our example problem, both matrices are 2 x 2s, so they can be multiplied in either order. There's one more requirement to check before you can take the inverse of a matrix. The determinant of the matrix must be nonzero. If the determinant is zero, the matrix does not have an inverse. Here's how to find the determinant in the simplest case, the 2 x 2 matrix: 2 x 2 matrix: The determinant of the matrix (abcd){\displaystyle {egin{pmatrix}a&b\c&d\end{pmatrix}}} is ad - bc. In other words, take the product of the main diagonal (top left to bottom right), then subtract the product of the anti-diagonal (top right to bottom left). For example, the matrix (7423){\displaystyle {egin{pmatrix}7&4\2&3\end{pmatrix}}} has the determinant (7)(3) - (4)(2) = 21 - 8 = 13. This is nonzero, so it is possible to find the inverse. If your matrix is 3 x 3 or larger, finding the determinant takes a bit more work: 3 x 3 matrix: Choose any element and cross out the row and column it belongs to. Find the determinant of the remaining 2 x 2 matrix, multiply by the chosen element, and refer to a matrix sign chart to determine the sign. Repeat this for the other two elements in the same row or column as the first one you chose, then sum all three determinants. Read this article for step-by-step instructions and tips to speed this up. Larger matrices: Using a graphing calculator or software is recommended. The method is similar to the 3 x 3 matrix method, but is tedious by hand. For example, to find the determinant of a 4 x 4 matrix, you need to find the determinants of four 3 x 3 matrices. If your matrix is not square, or if its determinant is zero, write "no unique solution." The problem is complete. If the matrix is square and its determinant is non-zero, continue to the next section for the next step: finding the inverse.
Understand matrix "division. Confirm the "divisor matrix" is square. Check that the two matrices can be multiplied together. Find the determinant of a 2 x 2 matrix. Find the determinant of a larger matrix. Continue on.