Learning Outcomes

A.

Perform the matrix operations of matrix addition, scalar multiplication, transposition and matrix multiplication. Identify when these operations are not defined. Represent these operations in terms of the entries of a matrix.

B.

Prove algebraic properties for matrix addition, scalar multiplication, transposition, and matrix multiplication. Apply these properties to manipulate an algebraic expression involving matrices.

C.

Compute the inverse of a matrix using row operations, and prove identities involving matrix inverses.

E.

Solve a linear system using matrix algebra.

F.

Use multiplication by an elementary matrix to apply row operations.

G.

Write a matrix as a product of elementary matrices.

You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix. It turns out that matrices are important not only for systems of equations but also in many applications.

Recall that a matrix is a rectangular array of numbers. Several of them are referred to as matrices. For example, here is a matrix.

Recall that the size or dimension of a matrix is defined as $m\times n$ where $m$ is the number of rows and $n$ is the number of columns. The above matrix is a $3\times 4$ matrix because there are three rows and four columns. You can remember the columns are like columns in a Greek temple. They stand upright while the rows lay flat like rows made by a tractor in a plowed field.

When specifying the size of a matrix, you always list the number of rows before the number of columns.You might remember that you always list the rows before the columns by using the phrase Rowman Catholic.

Definition C.1: Square Matrix

A matrix $A$ which has size $n\times n$ is called a square matrix . In other words, $A$ is a square matrix if it has the same number of rows and columns.

There is some notation specific to matrices which we now introduce. We denote the columns of a matrix $A$ by $A_j$ as follows

Therefore, $A_j$ is the $j^{th}$ column of $A$, when counted from left to right.

Definition C.2: The Zero Matrix

The $m\times n$ zero matrix is the $m\times n$ matrix having every entry equal to zero. It is denoted by $0.$

Example C.3: The Zero Matrix

The $2\times 3$ zero matrix is $0=\left[\begin{array}{ccc}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]$.

Definition C.4: Equality of Matrices

Let $A$ and $B$ be two $m\times n$ matrices. Then $A=B$ means that for $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right],$ $a_{ij}=b_{ij}$ for all $1\le i\le m$ and $1\le j\le n.$

In other words, two matrices are equal exactly when they are the same size and the corresponding entries are identical. Thus

because they are different sizes. Also,

because, although they are the same size, their corresponding entries are not identical.

In the following section, we explore addition of matrices.

When adding matrices, all matrices in the sum need have the same size. For example,

and

cannot be added, as one has size $3\times 2$ while the other has size $2\times 3$.

However, the addition

is possible.

Definition C.5: Addition of Matrices

Let $A=\left[a_{ij}\right]$ and $B=\left[b_{ij}\right]$ be two $m\times n$ matrices. Then $A+B=C$ where $C$ is the $m\times n$ matrix $C=\left[c_{ij}\right]$ defined by

Example C.6: Addition of Matrices of Same Size

Add the following matrices, if possible.

Solution

Recall that we use the word scalar when referring to numbers. Therefore, scalar multiplication of a matrix is the multiplication of a matrix by a number. To illustrate this concept, consider the following example in which a matrix is multiplied by the scalar $3$.

The new matrix is obtained by multiplying every entry of the original matrix by the given scalar.

Definition C.7: Scalar Multiplication of Matrices

If $A=\left[a_{ij}\right]$ and $k$ is a scalar, then $kA=\left[k{a}_{ij}\right].$

Example C.8: Effect of Multiplication by a Scalar

Find the result of multiplying the following matrix $A$ by $7$.

Solution

The next important matrix operation we will explore is multiplication of matrices. The operation of matrix multiplication is one of the most important and useful of the matrix operations. Throughout this section, we will also demonstrate how matrix multiplication relates to linear systems of equations.

Definition C.9: Row and Column Vectors

Matrices of size $n\times 1$ or $1\times n$ are called vectors. If $x$ is such a matrix, then we write $x_i$ to denote the entry of $x$ in the $i^{th}$ row of a column matrix, or the $i^{th}$ column of a row matrix.

The $n\times 1$ matrix

is called a column vector. The $1\times n$ matrix

is called a row vector.

We may simply use the term vector throughout this text to refer to either a column or row vector. If we do so, the context will make it clear which we are referring to.

Definition C.10: The Vector Form of a System of Linear Equations

Suppose we have a system of equations given by

We can express this system in vector form which is as follows:

Notice that each vector used here is one column from the corresponding augmented matrix. There is one vector for each variable in the system, along with the constant vector.

The first important form of matrix multiplication is multiplying a matrix by a vector. Consider the product given by

We will soon see that this equals

In general terms,

Thus you take $x_1$ times the first column, add to $x_2$ times the second column, and finally $x_3$ times the third column. The above sum is a linear combination of the columns of the matrix. When you multiply a matrix on the left by a vector on the right, the numbers making up the vector are just the scalars to be used in the linear combination of the columns as illustrated above.

Definition C.11: Multiplication of Vector by Matrix

Let $A=\left[a_{ij}\right]$ be an $m\times n$ matrix and let $x$ be an $n\times 1$ matrix given by

Then the product $Ax$ is the $m\times 1$ column vector which equals the following linear combination of the columns of $A$:

If we write the columns of $A$ in terms of their entries, they are of the form

Then, we can write the product $Ax$ as

Note that multiplication of an $m\times n$ matrix and an $n\times 1$ vector produces an $m\times 1$ vector.

Example C.12: A Vector Multiplied by a Matrix

Compute the product $Ax$ for

Solution

Definition C.13: The Matrix Form of a System of Linear Equations

Suppose we have a system of equations given by

Then we can express this system in matrix form as follows.

The expression $Ax=B$ is also known as the Matrix Form of the corresponding system of linear equations. The matrix $A$ is simply the coefficient matrix of the system, the vector $x$ is the column vector constructed from the variables of the system, and finally the vector $B$ is the column vector constructed from the constants of the system. It is important to note that any system of linear equations can be written in this form.

Notice that if we write a homogeneous system of equations in matrix form, it would have the form $Ax=0$, for the zero vector $0$.

You can see from this definition that a vector

will satisfy the equation $Ax=B$ only when the entries $x_1,x_2,\cdots ,x_n$ of the vector $x$ are solutions to the original system.

Definition C.14: Multiplication of Two Matrices

Let $A$ be an $m\times n$ matrix and let $B$ be an $n\times p$ matrix of the form

where $B_1,...,B_p$ are the $n\times 1$ columns of $B$. Then the $m\times p$ matrix $AB$ is defined as follows:

where ${(AB)}_k$ is an $m\times 1$ matrix or column vector which gives the $k^{th}$ column of $AB$.

Example C.15: Multiplying Two Matrices

Find $AB$ if possible.

Solution

Example C.16: Vector Times Vector Multiplication

Multiply if possible $\left[\begin{array}{c}\hfill 1\\ \hfill 2\\ \hfill 1\end{array}\right]\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill 1& \hfill 0\end{array}\right].$

Solution

Example C.17: A Multiplication Which is Not Defined

Find $BA$ if possible.

Solution

First check if it is possible. This product is of the form $\left(3\times 3\right)\left(2\times 3\right).$ The inside numbers do not match and so you can’t do this multiplication.

Example C.18: Multiplication by the Zero Matrix

Compute the product $A0$ for the matrix

and the $2\times 2$ zero matrix given by

Solution

In this product, we compute

Hence, $A0=0$.

Notice that we could also multiply $A$ by the $2\times 1$ zero vector given by $\left[\begin{array}{c}\hfill 0\\ \hfill 0\end{array}\right]$.The result would be the $2\times 1$ zero vector. Therefore, it is always the case that $A0=0$, for an appropriately sized zero matrix or vector.

In previous sections, we used the entries of a matrix to describe the action of matrix addition and scalar multiplication. We can also study matrix multiplication using the entries of matrices.

What is the $i{j}^{th}$ entry of $AB?$ It is the entry in the $i^{th}$ row and the $j^{th}$ column of the product $AB$.

Now if $A$ is $m\times n$ and $B$ is $n\times p$, then we know that the product $AB$ has the form

The $j^{th}$ column of $AB$ is of the form

which is an $m\times 1$ column vector. It is calculated by

Therefore, the $i{j}^{th}$ entry is the entry in row $i$ of this vector. This is computed by

Definition C.19: The $i{j}^{th}$ Entry of a Product

Let $A=\left[a_{ij}\right]$ be an $m\times n$ matrix and let $B=\left[b_{ij}\right]$ be an $n\times p$ matrix. Then $AB$ is an $m\times p$ matrix and the $\left(i,j\right)$-entry of $AB$ is defined as

Another way to write this is

In other words, to find the $\left(i,j\right)$-entry of the product $AB$, or ${(AB)}_{ij}$, you multiply the $i^{th}$ row of $A,$ on the left by the $j^{th}$ column of $B$. To express $AB$ in terms of its entries, we write $AB=\left[{(AB)}_{ij}\right]$.

Example C.20: The Entries of a Product

Example C.21: Finding the Entries of a Product

Determine if the product $AB$ is defined. If it is, find the $\left(2,1\right)$-entry of the product.

Solution

This product is of the form $\left(3\times 3\right)\left(3\times 2\right)$. The middle numbers match so the matrices are conformable and it is possible to compute the product.

Example C.22: Matrix Multiplication is Not Commutative

Compare the products $AB$ and $BA$, for matrices $A=\left[\begin{array}{cc}\hfill 1& \hfill 2\\ \hfill 3& \hfill 4\end{array}\right],B=\left[\begin{array}{cc}\hfill 0& \hfill 1\\ \hfill 1& \hfill 0\end{array}\right]$

Solution

First, notice that $A$ and $B$ are both of size $2\times 2$. Therefore, both products $AB$ and $BA$ are defined. The first product, $AB$ is

The second product, $BA$ is

Therefore, $AB\ne BA$.

This example illustrates that you cannot assume $AB=BA$ even when multiplication is defined in both orders. If for some matrices $A$ and $B$ it is true that $AB=BA$, then we say that $A$ and $B$ commute. This is one important property of matrix multiplication.

The following are other important properties of matrix multiplication. Notice that these properties hold only when the size of matrices are such that the products are defined.

Another important operation on matrices is that of taking the transpose. For a matrix $A$, we denote the transpose of $A$ by $A^T$. Before formally defining the transpose, we explore this operation on the following matrix.

What happened? The first column became the first row and the second column became the second row. Thus the $3\times 2$ matrix became a $2\times 3$ matrix. The number $4$ was in the first row and the second column and it ended up in the second row and first column.

Definition C.23: The Transpose of a Matrix

Let $A$ be an $m\times n$ matrix. Then $A^T$, the transpose of $A$, denotes the $n\times m$ matrix given by

The $\left(i,j\right)$-entry of $A$ becomes the $\left(j,i\right)$-entry of $A^T$.

Example C.24: The Transpose of a Matrix

Calculate $A^T$ for the following matrix

Solution

Definition C.25: Symmetric and Skew Symmetric Matrices

An $n\times n$ matrix $A$ is said to be symmetric if $A=A^T.$ It is said to be skew symmetric if $A=-A^T.$

Example C.26: Symmetric Matrices

Let

Example C.27: A Skew Symmetric Matrix

Let

Show that $A$ is skew symmetric.

Solution

There is a special matrix, denoted $I$, which is called to as the identity matrix. The identity matrix is always a square matrix, and it has the property that there are ones down the main diagonal and zeroes elsewhere. Here are some identity matrices of various sizes.

The first is the $1\times 1$ identity matrix, the second is the $2\times 2$ identity matrix, and so on. By extension, you can likely see what the $n\times n$ identity matrix would be. When it is necessary to distinguish which size of identity matrix is being discussed, we will use the notation $I_n$ for the $n\times n$ identity matrix.

The identity matrix is so important that there is a special symbol to denote the $i{j}^{th}$ entry of the identity matrix. This symbol is given by $I_{ij}={\delta }_{ij}$ where ${\delta }_{ij}$ is the Kronecker symbol defined by

$I_n$ is called the identity matrix because it is a multiplicative identity in the following sense.

Definition C.28: The Inverse of a Matrix

A square $n\times n$ matrix $A$ is said to have an inverse $A^{-1}$ if and only if

In this case, the matrix $A$ is called invertible.

Theorem C.29: Uniqueness of Inverse

Suppose $A$ is an $n\times n$ matrix such that an inverse $A^{-1}$ exists. Then there is only one such inverse matrix. That is, given any matrix $B$ such that $AB=BA=I$, $B=A^{-1}$.

Example C.30: Verifying the Inverse of a Matrix

Let $A=\left[\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 1& \hfill 2\end{array}\right].$ Show$\left[\begin{array}{cc}\hfill 2& \hfill -1\\ \hfill -1& \hfill 1\end{array}\right]$is the inverse of $A.$

Solution

To check this, multiply

and

showing that this matrix is indeed the inverse of $A.$

Example C.31: A Nonzero Matrix With No Inverse

Let $A=\left[\begin{array}{cc}\hfill 1& \hfill 1\\ \hfill 1& \hfill 1\end{array}\right].$ Show that $A$ does not have an inverse.

Solution

One might think $A$ would have an inverse because it does not equal zero. However, note that

If $A^{-1}$ existed, we would have the following

This says that

which is impossible! Therefore, $A$ does not have an inverse.

In the next section, we will explore how to find the inverse of a matrix, if it exists.