How to divide matrices (with pictures)

How to divide matrices (with pictures)
How to divide matrices (with pictures)
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If you know how to multiply two matrices together, then you should be able to divide one by the other. In fact, division is a language shortcut: it is more about multiplying one matrix by the inverse of the other. If we had to compare with the division between integers, it is as if we asked you to divide 10 by 5, you would then have to multiply 10 by the inverse of 5 (i.e. 5-1, that is to say 1/5). The operation is then: 10 x 5-1, and you should find the same result as directly dividing. The division of the two matrices being a mathematical nonsense, to multiply by the inverse of a matrix remains what comes closest to a division, without being one in the mathematical sense of the term. This operation is frequently used to solve systems of linear equations.

Steps

Part 1 of 3: Knowing if dividing a matrix is ​​possible

Divide Matrices Step 1

Step 1. Understand what “division” of one matrix by another is

In fact, it is an operation that does not exist as such: it makes no sense, mathematicians do not even use the term. The operation which comes closest to it is the multiplication of one matrix by the inverse of the other. In other words, you will never see the operation [A] ÷ [B], because it is mathematical nonsense. On the other hand, you can calculate [A] x [B]-1. If these two operations are equivalent for scalar quantities (numbers), you should use the second if you have to "divide" matrices.

  • The order is also important: the product [A] x [B]-1 is not equal to [B]-1 x [A]. To convince yourself, do both operations and you will see that the solutions are different.
  • So instead of writing (13263913) ÷ (7423) { displaystyle { begin {pmatrix} 13 & 26 \ 39 & 13 \ end {pmatrix}} div { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix} }}

    {begin{pmatrix}13&26\\39&13\end{pmatrix}}\div {begin{pmatrix}7&4\\2&3\end{pmatrix}}

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

    {begin{pmatrix}13&26\\39&13\end{pmatrix}}\times {begin{pmatrix}7&4\\2&3\end{pmatrix}}^{{-1}}

    Posez également l'opération suivante: (7423)−1×(13263913){displaystyle {begin{pmatrix}7&4\\2&3\end{pmatrix}}^{-1}\times {begin{pmatrix}13&26\\39&13\end{pmatrix}}}

    {begin{pmatrix}7&4\\2&3\end{pmatrix}}^{{-1}}\times {begin{pmatrix}13&26\\39&13\end{pmatrix}}

    , laquelle à une réponse différente de la première.

Divide Matrices Step 2

Step 2. Check that the divider die is square

The inverse of a matrix only exists for square matrices, that is, having the same number of rows and columns. If not, there would be several solutions to your problem.

  • The term "divider" is totally inappropriate, as it is by no means a division in the usual sense. In the operation [A] x [B]-1, the division matrix is ​​the matrix [B]. To return to our example, it is the matrix (7423) { displaystyle { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix}}}

    {begin{pmatrix}7&4\\2&3\end{pmatrix}}
  • Une matrice ayant un inverse est dite « inversible » ou « régulière », celle qui n'en a pas est dite « singulière ».
Divide Matrices Step 3

Step 3. See if the two matrices can be multiplied

To do this, check that the number of columns in the first matrix is ​​the same as the number of rows in the second. If this does not hold true for [A] x [B]-1 nor for [B]-1 x [A] is that the operation is not possible.

  • So, if [A] is a 4x3 matrix and [B] is a 2x2 matrix, your problem has no solution. The product [A] x [B]-1 cannot be done, because 4 ≠ 2, nor the product [B]-1 x [A], since 2 ≠ 3.
  • Observe that the inverse matrix [B]-1 has the same number of columns and rows as the starting matrix [B]. For now, it is not necessary to calculate it.
  • In the example chosen, the two matrices are of type 2 x 2, they can therefore be multiplied simply.
Divide Matrices Step 4

Step 4. Calculate the determinant of the division matrix

Indeed, before finding the inverse matrix, you must calculate the determinant of this matrix, which must be non-zero, otherwise you will not have an inverse: it is a sine qua non condition. The calculation of a determinant is quite simple.

  • For a 2 x 2 matrix, as is the case for the matrix (abcd) { displaystyle { begin {pmatrix} a & b \ c & d \ end {pmatrix}}}

    {begin{pmatrix}a&b\\c&d\end{pmatrix}}

    , le déterminant est: ad - bc. Vous faites le produit de la diagonale qui part du haut à gauche et va en bas à droite, produit auquel vous retranchez le produit de l'autre diagonale.

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

    a pour déterminant: (7)(3) - (4)(2) = 21 - 8 = 13. Comme il est non nul, il est possible d'établir la matrice inverse.

Divide Matrices Step 5

Step 5. Find the determinant of a larger matrix

If you are dealing with a 3 x 3 or larger matrix, the computation of the determinant is just a bit longer.

  • For a 3 x 3 matrix: take into account any element of the matrix and mentally remove the row and the column on which this element is located: you obtain a 2 x 2 matrix. Calculate the determinant according to the previous method. Do the same with a second element taken from the same column (or row), you have a second determinant. Finally, consider the third element of the column (or row), and you get a third determinant. Finally, add the three determinants. Check out this article to become an Ace at Matrix Determinants.
  • For the calculation of larger matrices, use a graphing calculator or calculation software. The method is the same as for the 3 x 3 matrix, but you understand that the calculations become, if not complicated, more numerous: the work quickly becomes tedious. To give you an idea of ​​the work to be done, finding the determinant of a 4 x 4 matrix is ​​to find the determinants of four 3 x 3 matrices.
Divide Matrices Step 6

Step 6. Now take stock

If your matrix is ​​not square or if the determinant is zero, your exercise stops there, because it does not admit a unique solution. If your matrix is ​​square and if its determinant is non-zero, then you can proceed to the next step: determining the inverse matrix.

Part 2 of 3: Find the inverse of a matrix

Divide Matrices Step 7

Step 1. Swap two elements of the main diagonal

With a 2 x 2 matrix, there is a trick that allows you to go faster. It consists in initially inverting the element located at the top left with the one located at the bottom right.

  • Let's take the example again: (7423) { displaystyle { begin {pmatrix} 7 & 4 \ 2 & 3 \ end {pmatrix}}}

    {begin{pmatrix}7&4\\2&3\end{pmatrix}}

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

    {begin{pmatrix}3&4\\2&7\end{pmatrix}}
  • Nota bene: Pour les matrices 3 x 3 ou plus grandes, les étudiants se servent de leur calculatrice scientifique, mais cela peut aussi se faire à la main, voyez plutôt à la fin de cette partie.
Divide Matrices Step 8

Step 2. Take the inverse of the other two elements

They do not move position, the operation therefore consists in multiplying the element at the top right and the element at the bottom left by -1.

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

    {begin{pmatrix}3&4\\2&7\end{pmatrix}}

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

    {begin{pmatrix}3&-4\\-2&7\end{pmatrix}}
Divide Matrices Step 9

Step 3. Find the inverse of the determinant

In the previous part, you have found the determinant of the matrix, it will be of use to us now. Write the reverse of this number in the form: 1de´terminant { displaystyle { frac {1} {d { acute {e}} ending}}}

{frac {1}{d{acute e}terminant}}
  • Dans notre exemple, le déterminant est 13. Son inverse est donc: 113{displaystyle {frac {1}{13}}}
  • {frac {1}{13}}
Divide Matrices Step 10

Step 4. Multiply this new matrix by the inverse of the determinant

Multiply each element of the matrix by the inverse you just found. The resulting matrix is ​​the inverse of the 2 x 2 matrix.

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

    {frac {1}{13}}\times {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}}}

    {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}
Divide Matrices Step 11

Step 5. Check your calculations

To do this, multiply the original matrix by the starting matrix. If your calculations are correct, you should find the identity matrix, either (1001) { displaystyle { begin {pmatrix} 1 & 0 \ 0 & 1 \ end {pmatrix}}}

{begin{pmatrix}1&0\\0&1\end{pmatrix}}

. Si c'est ce que vous constatez, vous pouvez passer au paragraphe suivant.

  • Reprenons notre exemple. Multipliez (313−413−213713)×(7423)=(1001){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}\times {begin{pmatrix}7&4\\2&3\end{pmatrix}}={begin{pmatrix}1&0\\0&1\end{pmatrix}}}
  • {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}\times {begin{pmatrix}7&4\\2&3\end{pmatrix}}={begin{pmatrix}1&0\\0&1\end{pmatrix}}
  • Pour vous rafraichir la mémoire sur la multiplication des matrices, lisez cet article.
  • Nota bene: la multiplication n'est pas commutative, ce qui signifie que l'ordre des opérants a son importance. Par contre, quand vous multipliez une matrice par son inverse, l'ordre importe peu, le résultat sera le même: la matrice identité.
Divide Matrices Step 12

Step 6. Review how to calculate the inverse of a 3 x 3 matrix

This article also applies to larger dies. Unless you want to understand how they are calculated, it's much easier and faster to use a graphing calculator (or dedicated software) for large matrices. If you need to do the math, see one of the possible methods below.

  • Join the identity matrix (I) to the right of your matrix. Thus, [B] → [B | I]. The identity matrix shows only "1" on the main diagonal and "0" on the other.
  • Start a series of linear cuts by dividing each item at the same place in the two halves by the same number. Continue in this way until the left half is the identity matrix.
  • Once the reductions are made, your matrix is ​​as follows: [I | B-1], I being the identity matrix. The right half of the extended matrix is ​​this time the inverse matrix (the one you are looking for) of the starting matrix.

Part 3 of 3: multiplying matrices to divide them

Divide Matrices Step 13

Step 1. Write the two possible equations

In classical algebraic calculus, multiplication is commutative. This is how: 2 x 6 = 6 x 2. On the other hand, it is not between matrices, which implies that you have two problems to solve.

  • In the equation x [B] = [A], the matrix x is equal to [A] x [B]-1.
  • In the equation [B] x = [A], the matrix x is equal to [B]-1 x [A].
  • If in a problem of linear equations, for example, you have to solve [A] = [C], and you are forced to multiply each member by [B]-1, [B]-1[TO] will not be equal to [C] [B]-1, because [B]-1 is to the left of [A] and that, in the other member, it is to the right to the right of [C]. You must respect the order.
Divide Step Matrices 14

Step 2. Determine the dimensions of the product matrix

These are the outer dimensions of the two matrices, which means that the produced matrix has as many rows as the first matrix and as many columns as the second matrix.

  • Let's go back to our original example. We had the matrices (13263913) { displaystyle { begin {pmatrix} 13 & 26 \ 39 & 13 \ end {pmatrix}}}

    {begin{pmatrix}13&26\\39&13\end{pmatrix}}

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

    {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}

    qui, toutes deux, sont des matrices 2 x 2, la réponse sera alors une matrice 2 x 2.

  • Il n'en va pas de même si, par exemple, la matrice [A] est une matrice

    Étape 4. x 3 et [B]-1, une matrice 3

    Étape 3.. Le produit [A] x [B]-1 donnera une matric

    Étape 4

    Étape 3..

Divide Matrices Step 15

Step 3. Find the value of the first element

To do this, read or reread this article or read the little memento that follows to refresh your memory.

  • To find the element of the first row and the first column of the matrix [A] [B]-1, you need to calculate the dot product of row 1 of matrix [A] by column 1 of matrix [B]-1, which for a 2 x 2 matrix gives: (a1, 1 × b1, 1) + (a1, 2 × b2, 1) { displaystyle (a_ {1, 1} times b_ {1, 1}) + (a_ {1, 2} times b_ {2, 1})}

    (a_{{1, 1}}\times b_{{1, 1}})+(a_{{1, 2}}\times b_{{2, 1}})
  • Dans notre exemple (13263913)×(313−413−213713){displaystyle {begin{pmatrix}13&26\\39&13\end{pmatrix}}\times {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}}
  • {begin{pmatrix}13&26\\39&13\end{pmatrix}}\times {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}

    , l'élément de la première ligne et de la première colonne sera:

    (13×313)+(26×−213){displaystyle (13\times {frac {3}{13}})+(26\times {frac {-2}{13}})}

    (13\times {frac {3}{13}})+(26\times {frac {-2}{13}})

    =(3)+(−4){displaystyle =(3)+(-4)}

    =(3)+(-4)

    =−1{displaystyle =-1}

    =-1
Divide Matrices Step 16

Step 4. Calculate all the dot products for each of the elements

Thus, the element located in position (2, 1) is the dot product of row 2 of matrix [A] by column 1 of matrix [B]-1. Try to do the following calculations on your own, you should find the following matrices:

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

    {begin{pmatrix}13&26\\39&13\end{pmatrix}}\times {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 vous devez calculer l'autre solution, elle est la suivante:(313−413−213713)×(13263913)=(−92193){displaystyle {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}\times {begin{pmatrix}13&26\\39&13\end{pmatrix}}={begin{pmatrix}-9&2\\19&3\end{pmatrix}}}
  • {begin{pmatrix}{frac {3}{13}}&{frac {-4}{13}}\\{frac {-2}{13}}&{frac {7}{13}}\end{pmatrix}}\times {begin{pmatrix}13&26\\39&13\end{pmatrix}}={begin{pmatrix}-9&2\\19&3\end{pmatrix}}

conseils

  • il est possible de diviser une matrice par un scalaire: divisez chaque élément par ce scalaire.

    • ainsi, si vous devez diviser la matrice (6824){displaystyle {begin{pmatrix}6&8\\2&4\end{pmatrix}}}
    • {begin{pmatrix}6&8\\2&4\end{pmatrix}}

      par 2, vous obtiendriez: (3412){displaystyle {begin{pmatrix}3&4\\1&2\end{pmatrix}}}

      {begin{pmatrix}3&4\\1&2\end{pmatrix}}

avertissements

  • une calculatrice ne peut faire que ce pour quoi elle est programmée, ce qui fait qu'elle n'est pas toujours cohérente concernant le calcul matriciel. ainsi, si vous voyez s'afficher à l'écran une réponse du type « 2e-8 », vous pouvez être certain que l'élément est en fait 0.

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