If you know how to multiply two matrices, you are well on your way to "dividing" one matrix by another. That word is in quotes because arrays cannot technically be split. Instead, one matrix is multiplied by the inverse of another. If this sounds strange, consider it in terms of some more familiar math: instead of calculating 10 ÷ 5, we can take the inverse of 5 (5-1 or 1/5), solve 10 x 5-1 and end with the same answer. That is why multiplying by the inverse of a matrix is considered the closest process to division defined in this branch of mathematics. These calculations are commonly used to solve systems of linear equations.
Quick summary
- There is no definition for matrix division. Instead, multiply the first matrix by the inverse of the second. Rewrite the problem [A] ÷ [B] as [A] * [B]-1 or [B]-1 * [TO].
- If the matrix [B] is not square or if its determinant is 0, write "there is no single solution". Otherwise, find the determinant of [B] and go to the next step.
- Find [B]-1 (the inverse of [B]).
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Multiply the matrices to find [A] * [B]-1 or [B]-1 * [TO]. Keep in mind that these will not necessarily give the same answer.
Steps
Part 1 of 3: Confirm that "splitting" is possible
Divide Matrices Step 1 Step 1. Understand matrix "division"
Technically, there is no matrix division. Dividing one matrix by another 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 if you used scalar quantities, 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 have to solve both to find all possible solutions.
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For example, instead of (13263913) ÷ (7423) { displaystyle { begin {pmatrix} 13 & 26 \ 39 & 13 \ end {pmatrix}} div { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix}} }
, escribe (13263913)∗(7423)−1{displaystyle {begin{pmatrix}13&26\\39&13\end{pmatrix}}*{begin{pmatrix}7&4\\2&3\end{pmatrix}}^{-1}}
Quizás también tengas que calcular (7423)−1∗(13263913){displaystyle {begin{pmatrix}7&4\\2&3\end{pmatrix}}^{-1}*{begin{pmatrix}13&26\\39&13\end{pmatrix}}}
, las cuales podrían tener una respuesta diferente.
Divide Matrices Step 2 Step 2. Confirm that the "dividing matrix" is square
To get the inverse of a matrix, it must be a square matrix with the same number of rows and columns. If the matrix you plan to invert is not square, there will be no single solution to the problem.
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The term "dividing matrix" is a bit vague as it is not technically a division problem. For [A] * [B]-1, this refers to the matrix [B]. In our example problem, this is (7423) { displaystyle { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix}}}
Divide Matrices Step 3 Step 3. Verify that both matrices can be multiplied
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. If this doesn't work in any of the provisions ([A] * [B]-1 or [B]-1 * [A]), there is no solution to the problem.
- For example, if [A] is a 4 x 3 matrix and [B] is a 2 x 2 matrix, there is no solution. [A] * [B]-1 does not work because 4 ≠ 2 and [B]-1 * [A] does not work because 2 ≠ 3.
- Note that the inverse [B]-1 it always has the same number of rows and columns as the original matrix [B]. You don't need to calculate the inverse to finish this step.
- In our example, both matrices are 2 x 2, so they can be multiplied in any order.
Divide Matrices Step 4 Step 4. Find the determinant of a 2 x 2 matrix
There is one more requirement that you must check before you can find the inverse of a matrix. The determinant of the matrix must not be zero. If the determinant is zero, the matrix does not have an inverse. This is how you can find the determinant in the simplest case, a 2 x 2 matrix:
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2 x 2 matrix:
the determinant of the matrix (abcd) { displaystyle { begin {pmatrix} a & b \ c & d \ end {pmatrix}}}
es ad - bc. En otras palabras, resta el producto de la diagonal principal (de la esquina superior izquierda hacia la esquina inferior derecha) al producto de la antidiagonal (de la esquina superior derecha hacia la esquina inferior izquierda).
- Por ejemplo, la matriz (7423){displaystyle {begin{pmatrix}7&4\\2&3\end{pmatrix}}}
tiene el determinante (7)(3) - (4)(2) = 21 - 8 = 13. Este no es cero, así que es posible encontrar la inversa.
- 3 x 3 matrix: choose any element and cross out the row and column to which it belongs. Find the determinant of the remaining 2 x 2 matrix, multiply by the chosen element, and consult a matrix sign chart to determine the sign. Repeat the procedure for the other two elements in the same row or column as the first one you have chosen and then add the three determinants. Read this article for step-by-step instructions and tips on how to speed up this process.
- Larger arrays: It is recommended to use a calculator or graphing software. The method is similar to the method for 3 x 3 matrices but is tedious to do by hand. For example, to find the determinant of a 4 x 4 matrix, you have to find the determinants of four 3 x 3 matrices.
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(7423) { displaystyle { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix}}}
→ (3427){displaystyle {begin{pmatrix}3&4\\2&7\end{pmatrix}}}
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Nota:
la mayoría de las personas usan calculadoras para encontrar la inversa de una matriz de 3 x 3 o más grande. Si quieres calcularlo a mano, consulta el final de esta sección.
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(3427) { displaystyle { begin {pmatrix} 3 & 4 \ 2 & 7 \ end {pmatrix}}}
→ (3−4−27){displaystyle {begin{pmatrix}3&-4\\-2&7\end{pmatrix}}}
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In our example, the determinant is 13. Its reciprocal is 113 { displaystyle { frac {1} {13}}}
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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}}}
- Para el problema de ejemplo, multiplica (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}}}
- Aquí puedes encontrar un curso de actualización sobre cómo multiplicar matrices.
- Nota: la multiplicación de matrices no es conmutativa, es decir, el orden de los factores importa. Sin embargo, al multiplicar una matriz por su inversa, ambas opciones darán como resultado la matriz identidad.
- Place the identity matrix I to the right of your matrix. For example, [B] → [B | I]. The identity matrix has "1" elements along the main diagonal and "0" elements in all other positions.
- Perform row operations to reduce the matrix until the left side is staggered, and then continue reducing until the left side is the identity matrix.
- Once you finish the operation, the matrix will be in the form [I | B-1]. In other words, the right hand side will be the inverse of the original matrix.
- [A] * [B]-1 is the solution of x to the problem x [B] = [A].
- [B]-1 * [A] is the solution of x to the problem [B] x = [A].
- If this is part of an equation, be sure to do the same on both sides. If [A] = [C], then [B]-1[TO] no equals [C] [B]-1 why]-1 it is to the left of [A] but to the right of [C].
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Going back to our original example, both (13263913) { displaystyle { begin {pmatrix} 13 & 26 \ 39 & 13 \ end {pmatrix}}}
como (313−413−213713){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}}
son matrices de 2 x 2, así que las dimensiones de la respuesta también serán de 2 x 2.
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Para usar un ejemplo más complicado, si [A] es una matriz de
Paso 4. x 3 y [B]-1 es una matriz de 3
Paso 3., las dimensiones de la matriz [A] * [B]-1 serán 4 x 3.
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To find row 1, column 1 of [A] [B]-1, find the dot product of row 1 of [A] and column 2 of [B]-1. 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}}
- En nuestro ejemplo, (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}}}
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(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}}}
- si necesitas encontrar la otra solución, (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}}}
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puedes dividir una matriz entre un escalar dividiendo cada elemento de la matriz entre el escalar.
- por ejemplo, la matriz (6824){displaystyle {begin{pmatrix}6&8\\2&4\end{pmatrix}}}
dividida entre 2 = (3412){displaystyle {begin{pmatrix}3&4\\1&2\end{pmatrix}}}
- las calculadoras no siempre son 100 % fiables cuando se trata de cálculos con matrices. por ejemplo, si tu calculadora te dice que un elemento es un número muy pequeño (2e-8, por ejemplo), lo más probable es que el valor sea cero.
Step 5. Find the determinant of a larger matrix
If the matrix is 3 x 3 or larger, finding the determinant takes a bit more work:
Step 6. Continue
If the matrix is not square or if the determinant is zero, write "there is no single solution." The problem is over. If the matrix is square and its determinant is not zero, move on to the next section for the next step: finding the inverse.
Part 2 of 3: Invert the Matrix
Step 1. Change the positions of the elements on the main diagonal of the 2 x 2 matrix
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 item in the upper left corner with the item in the lower right corner. For instance:
Step 2. Take the opposite side of the other two elements, but leave them in position
In other words, multiply the items in the upper right corner and the lower left corner by -1:
Step 3. Obtain the reciprocal of the determinant
You found the determinant of this matrix in the previous section so you don't have to calculate it again. Just write the reciprocal 1 / (determiner):
Step 4. Multiply the new matrix by the reciprocal of the determinant
Multiply each element of the new matrix by the reciprocal you just found. The resulting matrix is the inverse of the 2 x 2 matrix:
Step 5. Confirm that the inverse is correct
To check your work, multiply the inverse by the original matrix. If the inverse is correct, its product will always be the identity matrix, (1001) { displaystyle { begin {pmatrix} 1 & 0 \ 0 & 1 \ end {pmatrix}}}
. Si los cálculos dan resultado, pasa a la siguiente sección para terminar el problema.
Step 6. Review the matrix inversion for matrices 3 x 3 or larger
Unless you are learning this procedure for the first time, save time by using a graphing calculator or math software for larger matrices. If you need to calculate it by hand, this is a short summary of a method:
Part 3 of 3: Multiplying matrices to finish the problem
Step 1. Write both possible equations
In regular math with scalar quantities, multiplication is commutative: 2 x 6 = 6 x 2. This does not apply to matrices, so you may have to solve two problems:
Step 2. Find the dimensions of your 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.
Step 3. Find the value of the first element
Check out the article in the link for full instructions or refresh your memory with this summary:
, la fila 1 columna 1 de nuestra respuesta es:
(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 procedure for each position in the matrix
For example, the element in position 2, 1 is the dot product of row 2 of [A] and column 1 of [B]-1. Try to finish the example on your own. You should get the following responses: