2x3 Matrix Times 2x3 Matrix

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A system of an equation is a set of 2 or more equations, which have a shared gear up of unknowns and therefore a mutual solution. For linear equations, which graph equally straight lines, the mutual solution to a system is the point where the lines intersect. Matrices tin exist helpful for rewriting and solving linear systems.

i
Know your terminology. Linear equations have distinct components. The variable is the symbol (usually a letter like x or y) for a number you do non yet know. The constant is a number that remains consistent. The coefficient is a number before a variable, which is used to multiply it.^[1]
- For instance, in the linear equation 2x + 4y = 8, x and y are variables. The constant is viii. The numbers 2 and four are coefficients.
two

Recognize the class for a system of equations. A organization of equations with two variables tin be written as follows:ax + past = pcx + dy = qAny of the constants (p, q) tin can be nix, with the exception that each equation must have at least one variable (x, y) in it.

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three
Empathize matrix equations. When you lot accept a linear system, you can employ a matrix to rewrite information technology, and then apply the algebraic properties of that matrix to solve it. To rewrite a linear system, you use A to stand for the coefficients matrix, C to represent the constants matrix, and X to represent the unknown matrix.^[2]
- The linear arrangement to a higher place, for example, tin exist rewritten as a matrix equation as follows: A 10 Ten = C.
4
Empathize augmented matrices. An augmented matrix is a matrix obtained by appending columns of 2 matrices. If you have 2 matrices, A and C, which looks like this:

You tin create an augmented matrix past putting them together. The augmented matrix would look similar this:^[3]
- For example, consider the following linear system:
  2x + 4y = 8
  ten + y = two
  Your augmented matrix would exist a 2x3 matrix that looks similar this:

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one
Understand elementary operations. You lot tin can perform certain operations on a matrix to transform it while keeping information technology equivalent to the original. These are chosen elementary operations. To solve a 2x3 matrix, for example, y'all use elementary row operations to transform the matrix into a triangular one. Elementary operations include:^[4]
- swapping two rows.
- multiplying a row by a number different from zero.
- multiplying 1 row and so adding to another row.
two
Multiply the second row past a not-zero number. You desire to produce cipher in your 2nd row, so multiply in a way that lets you do that.^[v]
- For case, say y'all accept a matrix that looks like this:
  You tin can continue the first row and use information technology to produce zero in the second row. To do that, first multiply the second row by ii, as follows:
3
Multiply once again. In order to get to zip for the first row, you lot may demand to multiply once again, using the aforementioned principle.^[half-dozen]
- In the instance in a higher place, multiply the second row by -1, as follows:
  When you lot complete the multiplication, your new matrix looks similar this:
4
Add together the start row to the second row. Next, add the first and second rows to produce zero in the first column of the second row.
- In the example above, add the 2 rows together equally follows:
5
Write down the new linear system for the triangular matrix. At this point, you have a triangular matrix. Yous tin use that matrix to get a new linear organisation. The first column corresponds to the unknown x, and the second column corresponds to the unknown y. The third column corresponds to the costless member of an equation.^[vii]
- For the example higher up, your new system would therefore look like this:
6
Solve for i of the variables. Using your new system, determine which variable tin be adamant easily, and solve for it.
- In the case in a higher place, y'all'll want to "backsolve" – moving from the last equation to the first when solving for your unknowns. The second equation gives you lot an easy solution for y; since the x has been removed, y'all can encounter that y = 2.
7
Substitute to solve for the second variable. Once you've determined 1 of the variables, y'all can substitute its value into the other equation to solve for the other variable.
- In the example above, supervene upon the y with a two in the first equation to solve for x as follows:

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How do I summate the determinant of a 2x3 matrix?

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Community Answer

Determinants are defined for square matrices, and that definition doesn't utilise to non-squares like 2x3. You can compute the rank of any matrix to come across if its rows are linearly independent.

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The elements arrayed in a matrix are typically called "scalars."

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Remember that to solve a 2x3 matrix, you lot must stick to the elementary row operations. Y'all cannot use column operations.

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