How to Split Matrices (with Images)

Knowing how to multiply two matrices is already halfway to learning how to "divide" one matrix by another. The word "split" is written in quotation marks because arrays technically cannot be split. Instead, you need to multiply one matrix by the inverse of the other. If this sounds strange, consider this idea in terms of more common mathematical concepts: instead of calculating 10 ÷ 5, you can take the inverse of 5 (5-1 or 1/5), calculate 10 x 5-1 and get the same answer. Therefore, multiplication by the inverse of a matrix is ​​considered the closest process to division in this branch of mathematics. These calculations are commonly used to solve systems of linear equations.

Quick guide

1. There is no definition for matrix division. Instead, multiply the first matrix by the inverse of the second. Rewrite problem [A] ÷ [B] as [A] * [B]-1 or [B]-1 * [A].
2. If the matrix [B] is not square or if its determinant is equal to zero, write "there is no single solution". Otherwise, find the determinant of [B] and continue with the next step.
3. Calculate the value of [B]-1 (the inverse of [B]).
4. Multiply the matrices to calculate [A] * [B]-1 or [B]-1 * [A]. Be aware that this will not necessarily result in the same answer.

Steps

Part 1 of 3: Confirming that "Division" is Possible

Step 1. Understand the matrix "division"

Technically, such a concept does not exist. Dividing one matrix by another is an undefined function. The closest equivalent is multiplication by the inverse of another matrix. In other words, although [A] ÷ [B] is not defined, it is possible to calculate [A] * [B]-1. Since the two equations would be equivalent in scalar magnitude, this "looks" like matrix division, but it is important to use the correct terminology.

• Note that [A] * [B]-1 and [B]-1 *[A] are not the same problem. You may need to calculate both to find possible solutions.
• For example, instead of (13263913)÷(7423){displaystyle {begin{pmatrix}13&26\39&13\end{pmatrix}}\div {begin{pmatrix}7&4\2&3\end{pmatrix}} }

, escreva (13263913)∗(7423)−1{displaystyle {begin{pmatrix}13&26\\39&13\end{pmatrix}}*{begin{pmatrix}7&4\\2&3\end{pmatrix}}^{-1}}

Você também pode precisar calcular (7423)−1∗(13263913){displaystyle {begin{pmatrix}7&4\\2&3\end{pmatrix}}^{-1}*{begin{pmatrix}13&26\\39&13\end{pmatrix}}}

, que pode ter uma resposta diferente.

Step 2. Check that the "divider matrix" is square

To get the inverse of a matrix, it must be square, with the same number of rows and columns. Otherwise, there is no single solution to the problem.

• The term "dividing matrix" is a bit vague as, technically, it is not a division problem. To [A] * [B]-1, this is referring to the matrix [B]. In the example used, it is matrix (7423){displaystyle {begin{pmatrix}7&4\2&3\end{pmatrix}}}

Step 3. Check if two matrices can be multiplied together

To do this, the number of columns in the first matrix must equal the number of rows in the second matrix. If it doesn't work in any of the settings ([A] * [B]-1 or [B]-1 * [A]), so the problem has no solution.

• For example, if [A] is a 4 x 3 matrix and [B] is a 2 x 2 matrix, then there is no solution. [A] * [B]-1 cannot be calculated since 4 ≠ 2, and [B]-1 * [A] also not, since 2 ≠ 3.
• Note that the inverse of [B]-1 it always has the same number of rows and columns as the original matrix [B]. It is not necessary to calculate the inverse to complete this step.
• In the example used, both matrices are 2 x 2, so they can be multiplied in any order.

Step 4. Find the determinant of a 2 x 2 matrix

There is one more requirement to check before you can get the inverse of an array. Her determinant cannot be zero. Otherwise, the matrix will not have an inverse. See how to find the determinant in the simplest case, a 2 x 2 matrix:

• 2 x 2 matrix:

the determinant of (abcd){displaystyle {begin{pmatrix}a&b\c&d\end{pmatrix}}}

é ad - bc. Em outras palavras, pegue o produto da diagonal principal (do canto superior esquerdo para ao canto inferior direito), depois subtraia o produto da diagonal inversa (do canto superior direito para ao canto inferior esquerdo).

• Por exemplo, a matriz (7423){displaystyle {begin{pmatrix}7&4\\2&3\end{pmatrix}}}
• tem o determinante (7)(3) - (4)(2) = 21 - 8 = 13. Ele não é o número zero, então é possível encontrar o inverso.

Step 5. Find the determinant of a larger matrix

If the matrix is ​​3 x 3 or larger, a little more work is needed to find the determinant:

• 3 x 3 matrix: choose any element and cross out the row and column in which it belongs. Find the determinant of the remaining 2 x 2 matrix, multiply it by the chosen element, and look at the sign of the matrix graph to determine the sign. Repeat this step for the next two elements in the same row or column as the first chosen element, then add the three determinants together. Read this article for step-by-step instructions and tips on how to speed up this process.
• larger matrices: The use of a graphing calculator or software is recommended. The method is similar to the 3 x 3 matrix, but it takes longer to do 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.

Step 6. Continue

If the matrix is ​​not square, or if its determinant is equal to zero, write "there is no single solution". The problem is complete. If the matrix is ​​square and has a nonzero determinant, proceed to the next section to learn the next step: finding the inverse.

Part 2 of 3: Inverting an array

Step 1. Swap the positions of the 2 x 2 main diagonal elements

If the matrix is ​​2 x 2, you can use a shortcut to make this calculation much easier. The first step in this shortcut involves swapping the top left corner element with the bottom right corner element. For example:

• (7423){displaystyle {begin{pmatrix}7&4\2&3\end{pmatrix}}}

→ (3427){displaystyle {begin{pmatrix}3&4\\2&7\end{pmatrix}}}

• Observação:

a maioria das pessoas usa uma calculadora para encontrar o inverso de uma matriz 3 x 3 ou maior. Se quiser fazer o cálculo à mão, consulte o final da seção.

Step 2. Take the opposite of the other two elements, but leave them in position

In other words, multiply the elements in the upper "right" and lower "left" corners by -1:

• (3427){displaystyle {begin{pmatrix}3&4\2&7\end{pmatrix}}}

→ (3−4−27){displaystyle {begin{pmatrix}3&-4\\-2&7\end{pmatrix}}}

Step 3. Take the reciprocal of the determinant

You found the determinant of this matrix in the section above, so you don't have to do it again. Just write the reciprocal 1 / (determinant):

• In the example, the determinant is 13. Its reciprocal is 113{displaystyle {frac {1}{13}}}

Step 4. Multiply the new matrix by the reciprocal of the determinant

Multiply each element of the new matrix by the newly calculated reciprocal. The resulting matrix is ​​the inverse of the 2 x 2 matrix:

• 113∗(3−4−27){displaystyle {frac {1}{13}}*{begin{pmatrix}3&-4\-2&7\end{pmatrix}}}

=(313−413−213713){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}}

Step 5. Check if the inversion is correct

To do this, multiply the inverse by the original matrix. If the reverse is correct, the product will always be identical to the matrix, (1001){displaystyle {begin{pmatrix}1&0\0&1\end{pmatrix}}}

. Se estiver tudo certo, continue com a próxima seção para terminar o problema.

• Para o exemplo utilizado, multiplique (313−413−213713)∗(7423)=(1001){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}*{begin{pmatrix}7&4\\2&3\end{pmatrix}}={begin{pmatrix}1&0\\0&1\end{pmatrix}}}
• Veja o artigo Como Multiplicar Matrizes caso precise de ajuda.
• Observação: a multiplicação de matriz não é comutativa, ou seja, a ordem dos fatores influencia no resultado. No entanto, ao multiplicar uma matriz pelo seu inverso, ambas as opções vão resultar na matriz identidade.

Step 6. See in this article how to invert a 3x3 or larger matrix

Unless you are learning this process for the first time, save time by using a graphing calculator or math software to do math with larger matrices. If you don't need to do the calculation by hand, see a quick guide to a method:

• Place the identity matrix I to the right of your matrix. For example, [B] → [B | I]. The identity matrix has a "1" element along with the main diagonal and a "0" element without all other positions.
• Perform inline operations to reduce the matrix until the left side is in scaled form, then continue the reduction until the left side is identical to the identity matrix.
• At the end of the operation, the matrix will be in the form [I | B-1]. In other words, the right side is going to be the inverse of the original matrix.

Part 3 of 3: Multiplying the matrices to complete the problem

Step 1. Write the two possible equations

In "common mathematics" with scalar quantities, multiplication is commutative; 2 x 6 = 6 x 2. However, the same is not true for matrices, so you need to calculate two problems:

• [A] * [B]-1 is the solution x to problem x [B] = [A].
• [B]-1 * [A] is the solution x to problem [B] x = [A].
• If this is part of an equation, perform the same operation on both sides. If [A] = [C], then [B]-1[THE] not is equal to [C][B]-1, because [B]-1 it is on the left side of [A], but on the right side of [C].

Step 2. Find the dimensions of the answer

The dimensions of the final matrix are the external dimensions of the two factors. It has the same number of rows as the first matrix, and the same number of columns as the second matrix.

• Returning to the original problem, both (13263913){displaystyle {begin{pmatrix}13&26\39&13\end{pmatrix}}}

quanto (313−413−213713){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}}

são matrizes 2 x 2, então as dimensões da resposta também serão 2 x 2.

• Para usar um exemplo mais complicado, se [A] é uma matriz

Passo 4. x 3 e [B]-1 é uma matriz 3

Passo 3., então a matriz [A] * [B]-1 possui dimensões 4 x 3.

Step 3. Calculate the value of the first element.

See the article linked above for more detailed instructions, or refresh your memory with the following summary:

• To find row 1 and column 1 of [A][B]-1, find the dot product of row [A] 1 and column [B]-1 2. That is, for a 2 x 2 matrix, calculate a1, 1∗b1, 1+a1, 2∗b2, 1{displaystyle a_{1, 1}*b_{1, 1}+a_{1, 2 }*b_{2, 1}}

• No exemplo utilizado (13263913)∗(313−413−213713){displaystyle {begin{pmatrix}13&26\\39&13\end{pmatrix}}*{begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}}
• , a linha 1 coluna 1 da resposta é:

(13∗313)+(26∗−213){displaystyle (13*{frac {3}{13}})+(26*{frac {-2}{13}})}

=3+−4{displaystyle =3+-4}

=−1{displaystyle =-1}

Step 4. Repeat the dot product process for each position in the matrix

For example, the element at position 2, 1 is the dot product of row [A] 2 and column [B]-1 1. Try to complete the example yourself. You should get the following answers:

• (13263913)∗(313−413−213713)=(−1107−5){displaystyle {begin{pmatrix}13&26\39&13\end{pmatrix}}*{begin{pmatrix}{frac {3} {13}}&{frac {-4}{13}}\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}={begin {pmatrix}-1&10\7&-5\end{pmatrix}}}

• caso precise encontrar outra solução, (313−413−213713)∗(13263913)=(−92193){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}*{begin{pmatrix}13&26\\39&13\end{pmatrix}}={begin{pmatrix}-9&2\\19&3\end{pmatrix}}}

dicas

• você pode dividir uma matriz por uma grandeza escalar dividindo cada elemento da matriz pela grandeza.

• por exemplo, a matriz (6824){displaystyle {begin{pmatrix}6&8\\2&4\end{pmatrix}}}
• dividida por 2 = (3412){displaystyle {begin{pmatrix}3&4\\1&2\end{pmatrix}}}

avisos

• as calculadoras nem sempre são 100% precisas no que diz respeito aos cálculos de matrizes. por exemplo, se a calculadora informa que um elemento é um número muito pequeno (2e-8, por exemplo), é provável que o valor seja zero.