Compute Rank Of Nonsquare Matrix
If the coefficient matrix is rectangular, linsolve returns the rank of the coefficient matrix as the second output argument. Show this behavior.
syms a b x yA = B = = linsolve
Warning: Solution is not unique because the system is rank-deficient. In sym.linsolve at 67 X = x/a -/ 0R =2
Just For Fun Do It Again
For fun , let us do this all again, but put matrix “X” first.
I want to show you this way, because many people think the solution above is so neat it must be the only way.
So we will solve it like this:
XA = B
And because of the way that matrices are multiplied we need to set up the matrices differently now. The rows and columns have to be switched over :
And XA = B looks like this:
Using Matrices When Solving System Of Equations
Matrices could be used to solve systems of equations but first one must master to find the inverse of a matrice, C-1.
A matrices C will have an inverse C-1 if and only if the determinant of C is not equal to zero.
$$if\: c=\begin a & b\\ c & d \end \ \ \ \begin a & b\\ c & d \end\neq 0\\ \\ \\ then\: C^=\frac\begin d & -b\\ -c & a \end$$
We will now in an example show how to solve systems of equations using matrices and the inverse of matrices.
Example
Consider the following simultaneous equations
$$\left\ 3x+y=5\\ 2x-y=0\\ \end\right.$$
Provided that we know how to multiply matrices we realize that our equations could be written as
$$\begin 3 & 1\\ 2 & -1 \end\cdot \begin x\\ y\\ \end=\begin 5\\ 0 \end$$
First we find the inverse of the coefficient matrix:
$$C^=\frac\begin -1 & -1\\ -2& 3 \end=$$
$$=-\frac\begin -1 & -1\\ -2& 3 \end$$
The next step is to multiply both sides of our matrix equation by the inverse matrix:
$$-\frac\begin -1 & -1\\ -2& 3 \end \begin 3 & 1\\ 2 & -1 \end\cdot \begin x\\ y\\ \end= -\frac\begin -1 & -1\\ -2& 3 \end \begin 5\\ 0 \end$$
$$-\frac\begin -5 & 0\\ 0 & -5 \end\cdot \begin x\\ y \end=-\frac\begin -5\\ -10 \end$$
$$\begin 1 & 0\\ 0 & 1 \end\cdot \begin x\\ y \end=\begin 1\\ 2 \end$$
Our solution is , the easiest way to check if we are right is to plug our values into our original equations.
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How To Solve Matrix Equations
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In linear algebra, matrix equations are very similar to normal algebraic equations, in that we manipulate the equation using operations to isolate our variable. However, the properties of matrices restrict a few of these operations, so we have to ensure that every operation is justified.
The most important property of a matrix when dealing with matrix equations is the invertibility of a matrix. Therefore, we will begin by reviewing the relevant theorems.
Multiplying A Matrix By Another Matrix
But to multiply a matrix we need to do the “dot product” of rows and columns … what does that mean? Let us see with an example:
To work out the answer for the 1st row and 1st column:
The “Dot Product” is where we multiply matching members, then sum up:
= 1×7 + 2×9 + 3×11 = 58
We match the 1st members , multiply them, likewise for the 2nd members and the 3rd members , and finally sum them up.
Want to see another example? Here it is for the 1st row and 2nd column:
= 1×8 + 2×10 + 3×12 = 64
We can do the same thing for the 2nd row and 1st column:
= 4×7 + 5×9 + 6×11 = 139
And for the 2nd row and 2nd column:
= 4×8 + 5×10 + 6×12 = 154
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Example: The Local Shop Sells 3 Types Of Pies
- Apple pies cost $3 each
- Cherry pies cost $4 each
- Blueberry pies cost $2 each
And this is how many they sold in 4 days:
Now think about this … the value of sales for Monday is calculated this way:
So it is, in fact, the “dot product” of prices and how many were sold:
= $3×13 + $4×8 + $2×6 = $83
And here is the full result in Matrix form:
They sold $83 worth of pies on Monday, $63 on Tuesday, etc.
Another Way To Think About It
Think about what a matrix represents. This very simple matrix could represent 5x + 2y + 5z. And this matrix could equal 2x + y + 6z. If you add them together using algebra, you would get:5x + 2y + 5z + 2x + y + 6z = 7x + 3y + 11z.This is the same result as you would get from adding the entries in the matrices together.
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What Is The Inverse Of A Matrix
Just like a number has a reciprocal …
Reciprocal of a Number
matrix inverse :
Inverse of a Matrix
1A
And there are other similarities:
When we multiply a number by its reciprocal we get 1:
181
When we multiply a matrix by its inverse we get the Identity Matrix :
I
Same thing when the inverse comes first:
181
We just mentioned the “Identity Matrix”. It is the matrix equivalent of the number “1”:
- It is “square” ,
- It has 1s on the diagonal and 0s everywhere else.
- Its symbol is the capital letter I.
The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc …
What Is The Determinant Of A Matrix
The determinant of a matrix is just a special number that is used to describe matrices for finding solutions to systems of linear equations, finding inverse matrices and for various applications in calculus. A definition in plain English is impossible to pin down its usually defined in mathematical terms or in terms of what it can help you do. The determinant of a matrix has several properties:
- It is a real number. This includes negative numbers.
- Determinants only exist for square matrices.
- An inverse matrix only exists for matrices with non-zero determinants.
The symbol for the determinant of a matrix A is |A|, which is also the same symbol used for absolute value, although the two have nothing to do with each other.
The formula for calculating the determinant of a matrix differs according to the size of the matrix.
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How To Operate With Matrices
Matrices are considered equal if they have the same dimensions and if each element of one matrix is equal to the corresponding element of the other matrix. You may multiply a matrix by any constant, this is called scalar multiplication.
Example
$$2\begin 1 & 2\\ 3 & 4 \end= \begin 2}\cdot 1 & 2}\cdot2\\ 2}\cdot3 & 2}\cdot4 \end = \begin 2 & 4\\ 6 & 8 \end$$
Matrices of the same size can be added and subtracted entry wise and matrices of compatible sizes can be multiplied.
Example
Add the two matrices A and B:
$$A=\begin 2} & -1}\\ 1} & 0} \endB=\begin 1} & 4} \\ 2} & \, 3} \end$$
This is possible since A and B since both matrices have two rows and two columns. We add each element in matrix A to the corresponding element in matrix B:
$$A+B=\begin 2}+ 1} & -1}+ 4} \\ 1}+ 2} & 0}+\, 3} \end=\begin 3 & 3\\ 3 & 3 \end$$
One subtracts matrices in the same way.
If you multiply the matrix Ap*q and the matrix Bm*n the product is
$$A_+B_=_$$
The element in the g-th row and the h-th column of AB is the sum of the product of the corresponding elements in the g-th row of A and the h-th column of B. You may only multiply two matrices if the number of columns in the first matrix and the number of rows in the second are equal.
Example
Multiply the matrices A and B:
$$A=\begin 1} & 3}\\ -1} & 0} \endB=\begin 2} & 1} & 1}\\ -1} & 2}& 4} \end$$
This is possible since the first matrix contains 2 columns and the second contains 2 rows.
What Is Matrix Equation
A matrix equation is of the form AX = B where A represents the coefficient matrix, X represents the column matrix of variables, and B represents the column matrix of the constants that are on the right side of the equations in a system. Let us consider a system of n nonhomogenous equations in n variables.
a x + a x + … + a x = b
a x + a x + … + a x = b
….
a x + a x + … + a x = b
Then the matrix equation that corresponds to the above system is:
AX = B, where
- A = A matrix made of coefficients = \
- X = Column matrix of variables = \(\left[\beginx_ \\
- B = Column matrix of constants = \(\left[\beginb_ \\
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What Is A Transpose Matrix
A transpose matrix is just where you switch all of the rows of the matrix into columns. Transpose matrices are useful in complex multiplication.
An alternate way of describing a transpose matrix is that an element at row r and column c is transposed to row c and column r. For example, an element in row 2, column 3 would be transposed to column 2, row 3. The dimension of the matrix also changes. For example, if you had a 4 x 5 matrix you would transpose to a 5 x 4 matrix.
A symmetric matrix is a special case of a transpose matrix it is equal to its transpose matrix.In more formal terms, A = AT.
Definition Of A Singular Matrix
A quick look at a matrix can possibly tell you if it is a singular matrix. If the matrix is square and has one row or column of zeros or two equal columns or two equal rows, then its a singular matrix. For example, the following ten matrices are all singular :
There are other types of singular matrices, some are not quite so easy to spot. Therefore, a more formal definition is necessary.
The following three properties define a singular matrix:
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A Real Life Example: Bus And Train
A group took a trip on a bus, at $3 per child and $3.20 per adult for a total of $118.40.
They took the train back at $3.50 per child and $3.60 per adult for a total of $135.20.
How many children, and how many adults?
First, let us set up the matrices :
This is just like the example above:
XA = B
So to solve it we need the inverse of “A”:
2424? That equals 0, and 1/0 is undefined.We cannot go any further! This matrix has no Inverse.
Such a matrix is called “Singular”,which only happens when the determinant is zero.
And it makes sense … look at the numbers: the second row is just double the first row, and does not add any new information.
And the determinant 2424 lets us know this fact.
Produce Equivalent Matrices Using Elementary Row Operations
Since the matrix is essentially the coefficients and constants of a linear system, the three row operations preserve the matrix. For example, swapping two rows simply means switching their position within the matrix. Also, when solving a system of linear equations by the elimination method, row multiplication would be the same as multiplying the whole equation by a number to obtain additive inverses so that a variable cancels. Finally, row addition is also the same as the elimination method, when one chooses to add or subtract the like terms of the equations to obtain the variable. Therefore, row operations preserve the matrix and can be used as an alternative method to solve a system of equations.
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Matrix Addition: More Examples
Matrix addition is just a series of additions. For a 2×2 matrix:
- Add the top left numbers together and write the sum in a new matrix, in the top left position.
- Add the top right numbers together and write the sum in the top right.
- Add the bottom left numbers together and write the sum in the bottom left.
- Add the bottom right numbers together and write the sum in the bottom right:
Use exactly the same procedure for a 2×3 matrix:
In fact, you can use this basic technique for any matrix addition as long as your matrices have the same dimensions . In other words, if the matrices are the same size, you can add them. If they arent the same size, you cant add them.
- A matrix with 4 rows and 2 columns can be added to a matrix with 4 rows and 2 columns.
- A matrix with 4 rows and 2 columns cannot be added to a matrix with 5 rows and 2 columns.
The above technique is sometimes called the entrywise sum as youre simply adding entries together and noting the result.
Imagine We Can’t Divide By Numbers
… and someone asks “How do I share 10 apples with 2 people?”
But we can take the reciprocal of 2 , so we answer:
10 × 0.5 = 5
They get 5 apples each.
The same thing can be done with matrices:
Say we want to find matrix X, and we know matrix A and B:
XA = B
It would be nice to divide both sides by A , but remember we can’t divide.
But what if we multiply both sides by A-1 ?
XAA-1 = BA-1
And we know that AA-1 = I, so:
XI = BA-1
We can remove I :
X = BA-1
And we have our answer
In that example we were very careful to get the multiplications correct, because with matrices the order of multiplication matters. AB is almost never equal to BA.
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Adding And Subtracting Matrices
Matrix addition and subtraction operate element-by-element. The two input matrices must have the same dimensions. The result is a new matrix of the same dimensions where each element is the sum or difference of each corresponding input element. For example, consider combining portfolios of different quantities of the same stocks .
Portfolios_PQ = Portfolios_RS = NewPortfolios = Portfolios_PQ + Portfolios_RS
NewPortfolios = 275 325 700 600 400 650
Adding or subtracting a scalar and a matrix is allowed and also operates element-by-element.
SmallerPortf = NewPortfolios-10
SmallerPortf = 265.00 315.00 690.00 590.00 390.00 640.00
How To Write A System Of Equations As A Matrix Equation
We can write a system of equations as a matrix equation AX = B. Here are the steps for the same:
- Maintain the order of the variables to be the same in all the equations.
- Get all variables to the left side and send the constants to the right side of every equation.
- Write the matrix A with all the coefficients of variables, where each row of A represents the coefficients of variables in one equation.
- Write the matrix X with variables in order as a column.
- Write the matrix B with constants in order as a column.
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Properties Of Transpose Matrices
Properties for transpose matrices are similar to the basic number properties that you encountered in basic algebra . The basic properties for matrices are:
- T = A: the transpose of a transpose matrix is the original matrix.
- T = AT + BT: The transpose of two matrices added together is the same as the transpose of each individual matrix added together.
- T = rAT: when a matrix is multiplied by a scalar element, it doesnt matter which order you transpose in .
- T = BT AT: the transpose of two matrices multiplied together is the same as the product of their transpose matrices in reverse order.
- T = -1: the transpose and the inverse of a matrix can be performed in any order.
Evaluate The Determinant Of A 2 2 Matrix
If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix [ we first write it as | | . To get the real number value of the determinate we subtract the products of the diagonals, as shown.
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Eigen Values And Eigen Vectors Of Matrices
If A is any square matrix of order ‘n’, a matrix of A – I can be formed, where I is a unit matrix of order n, such that the number , called the eigenvalue and a non-zero vector v, called the eigenvector, satisfy the equation, Av = v. is an eigenvalue of an n×n-matrix A if and only if A In is not invertible, which is equivalent to Det = 0.
Row Reduction: Solving A System Of Linear Equations
In row reduction, the linear system:
\displaystyle x+3y-2z=5 \\ 3x+5y+6z=7 \\ 2x+4y+3z=8
Is represented as an augmented matrix:
\displaystyle A=\begin 1 & 3 & -2 & 5 \\ 3 & 5 & 6 & 7 \\ 2 & 4 & 3 & 8 \end
This matrix is then modified using elementary row operations until it reaches reduced row echelon form.
Because these operations are reversible, the augmented matrix produced always represents a linear system that is equivalent to the original.
There are several specific algorithms to row-reduce an augmented matrix, the simplest of which are Gaussian elimination and Gauss-Jordan elimination. This computation can be done by hand or on the calculator under the matrix function rref .
The final matrix is in reduced row echelon form, and represents the system x=-15, y=8z=2.
\displaystyle A=\begin 1 & 0 & 0 & -15 \\ 0 & 1 & 0 & 8 \\ 0 & 0 & 1 & 2 \end
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