In this explainer, we will learn how to combine the operations of addition, subtraction, scalar multiplication, and transposing matrices.
Let us begin by recalling the notation for a matrix that is useful for defining matrix operations. If we denote for integers and to be a sequence of numbers, then denotes the matrix whose entry in the row and column is given by . If the order of the matrix is clear from the context, we can omit from the notation to write .
Using this notation, we will recall a few matrix operations to be used in this explainer. At the simplest level, two matrices of the same order can be combined with two of the most familiar mathematical tools: addition and subtraction of matrices.
Definition: Matrix Addition and Subtraction
Given a pair of matrices of the same order, we can add or subtract these matrices by adding or subtracting each corresponding entry. In other words, let and . Then, where and satisfying
There is also scalar multiplication to a matrix, which mimics our understanding of the multiplication of real numbers.
Definition: Scalar Multiplication of Matrices
Given a scalar and a matrix, the scalar multiplication of a matrix is computed by multiplying the scalar by each entry of the matrix. In other words, let and be a scalar. Then, where with
When we combine matrix addition, subtraction, and scalar multiplication, we should be cautious to follow the correct order of operations. The order of matrix operations is analogous to the order of operations in real numbers.
How To: Ordering Matrix Operations Involving Matrix Addition, Subtraction, and Scalar Multiplication
A combination of matrix operations involving matrix addition, subtraction, and scalar multiplication is evaluated in the following order:
- Compute the operation within parentheses.
- Compute the scalar multiplication.
- Compute the matrix addition.
- Compute the matrix subtraction.
We can see that the order of matrix operations involving these operations is analogous to that of real number operations, which is often known as PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction). The operations listed above only contain parentheses, multiplication, addition, and subtraction.
In our first example, we will combine matrix addition and scalar multiplication.
Example 1: Finding the Sum of Two Matrices with Scalar Multiplication
Given that what is ?
Answer
Recall that when combining matrix operations, we must first compute the expression inside the parentheses. Hence, to compute , we need to first find the matrix addition . Recall that we can add a pair of matrices of the same order by adding each corresponding entry in the matrices. We can see that both and are matrices, which means that they are of the same order. Hence, the addition is well defined and is given by
Now that we have obtained matrix , we can multiply this matrix by the scalar to finish our computation. Recall that scalar multiplication to a matrix is performed by multiplying the scalar by each entry of the matrix. This gives us
Hence,
Let us consider another example of combining matrix addition, subtraction, and scalar multiplication by following the correct order of operations.
Example 2: Applying Operations on Matrices Involving Scalar Multiplication to Find an Unknown Matrix
Given that what is the matrix ?
Answer
In this example, we need to compute an expression that involves scalar multiplication, addition, and subtraction of matrices. Recall that a combination of matrix operations involving matrix addition, subtraction, and scalar multiplication is evaluated in the following order:
- Compute the operation within parentheses.
- Compute the scalar multiplication.
- Compute the matrix addition.
- Compute the matrix subtraction.
Since the given expression in our example does not involve any parentheses, we begin by finding the scalar multiplications and . Recall that scalar multiplication to a matrix is performed by multiplying the scalar by each entry of the matrix. This gives us
Now that we have computed the scalar multiplication, we can consider matrix addition in the expression . The addition in this expression is . Recall that we can add or subtract a pair of matrices of the same order by adding or subtracting each corresponding entry in the matrices. We can see that the matrices and are matrices, which means that the matrix addition is well defined and is given by
Finally, we can finish the computation by considering matrix subtraction. We can see that and are both matrices. Hence, the subtraction is well defined and is given by
Hence,
In the next example, we will solve a matrix equation to find an unknown matrix in an equation.
Example 3: Finding an Unknown Matrix in an Equation
Fill in the blank: If , then .
Answer
In this example, we need to find the unknown matrix from the given equation. Recall that , which is on the right-hand side of the given equation, is the zero matrix of certain order.
The left-hand side of the equation is given by the addition of two matrices. We know that we can only add matrices of the same order, and the second matrix in the addition is a matrix. Hence, matrix must also be a matrix. Matrix is obtained by scalar multiplication to matrix . Since scalar multiplication does not change the order of a matrix, matrix is also of order . We know that adding two matrices of the same order preserves the order of the matrix. This means that the zero matrix on the right-hand side of the equation is a matrix.
We can find the unknown matrix using two different methods, as we will demonstrate below.
Method 1
We can denote to write
On the left-hand side of this equation, we have scalar multiplication and matrix addition. We recall that scalar multiplication should be computed before matrix addition, so let us first find the scalar multiplication . Recall that scalar multiplication to a matrix is performed by multiplying the scalar by each entry of the matrix. This gives us
Substituting this expression into our equation gives
Next, we need to compute the matrix addition on the left-hand side of this equation. Recall that we can add a pair of matrices of the same order by adding each corresponding entry in the matrices. We can see that both matrices are of order , so the addition is well defined and is given by
Substituting this expression on the left-hand side of the equation gives
Recall that two matrices of the same order are equal to each other if the corresponding entries in each matrix are equal. This gives us four equations:
We can rearrange each equation to obtain
This gives all entries of , which means
Method 2
Alternatively, we can solve the matrix equation by applying matrix operations directly to the equation so that the left-hand side of the equation is equal to . We begin with the equation
We want to isolate on the left-hand side of the equation. So, we subtract the second matrix from both sides of the equation to write
Subtracting a matrix from itself produces a zero matrix. We can also consider the right-hand side of the equation as adding the zero matrix and multiplied by the other matrix. We know that a zero matrix is an additive identity in matrix addition, which means that adding a matrix to a zero matrix does not change the matrix. Hence, we can write this equation as
Next, we can multiply both sides of the equation by the scalar to obtain
Finally, we need to find the scalar multiplication on the right-hand side of the equation:
This is option B.
In the previous example, we solved a matrix equation involving matrix addition, subtraction, and scalar multiplication. These three operations of matrices are analogous to the corresponding operations of real numbers. We can observe that the second method of solving this matrix equation is very similar to the method of solving an analogous real-number equation.
Transposition is a completely new concept for matrices that does not exist for real numbers, and we can combine this operation with the addition, subtraction, and scalar multiplication of matrices to produce problems that are unique to matrix operations.
Definition: Matrix Transposition
If we transpose an matrix, we obtain an matrix. We can obtain the transpose of a matrix by writing the rows of the matrix as the corresponding columns. Using matrix notation, we can write
Where does the transposition fit into the order of matrix operations? We can observe that the transposition of a matrix “looks like” an exponent of a matrix. While transposition is not analogous to taking an exponent, the order of transposition fits in analogously as the order of taking an exponent in real-number operations. Thinking of PEMDAS, we can see that the term “exponents,” hence transposition, is between parentheses and multiplication.
How To: Ordering Matrix Operations Involving Matrix Addition, Subtraction, Scalar Multiplication, and Transposition
A combination of matrix operations involving matrix addition, subtraction, scalar multiplication, and transposition is evaluated in the following order:
- Compute the operation within parentheses.
- Compute the transposition.
- Compute the scalar multiplication.
- Compute the matrix addition.
- Compute the matrix subtraction.
In the next example, we will combine the transposition and subtraction of matrices.
Example 4: Combining Matrix Transposition and Subtraction
Fill in the blank: .
Answer
In this example, we need to compute the matrix subtraction and transposition. Recall that the matrix transposition should be computed before the matrix subtraction. So, let us begin by finding the transpose of the second matrix on the left-hand side of the equation. Recall that we can obtain the transpose of a matrix by writing the rows of the matrix as the corresponding columns. This leads to
Next, let us consider the matrix subtraction:
Recall that we can subtract a pair of matrices of the same order by subtracting each corresponding entry in the matrices. We can see that both matrices are matrices, which means that the subtraction is well defined. Subtracting corresponding entries, we obtain
This completes the stated matrix operations. We can see that the resulting matrix is a diagonal matrix with 1 in each diagonal entry. Recall that such a matrix is an identity matrix and is denoted by .
This is option D.
In the next example, we will find an unknown matrix in an equation that involves matrix addition and transposition.
Example 5: Finding an Unknown Matrix in an Equation
Solve for matrix in the matrix equation , where is a unit matrix with size and .
Answer
In this example, we need to find an unknown matrix in a given equation. We can apply matrix operations directly to the equation to find the unknown matrix .
Let us begin by finding the transpose on the left-hand side of the given equation. Recall that we can obtain the transpose of a matrix by writing the rows of the matrix as the corresponding columns. This leads to
Next, let us find the matrix on the right-hand side of the equation. We are given that is a unit matrix, which is also known as an identity matrix. Recall that an identity, or unit, matrix is a diagonal matrix with 1s in the diagonal entries. This means
We can compute the scalar multiplication by multiplying the scalar 2 by each entry of the matrix. This gives us
Then, we can write the given equation as
Next, we add the matrix to both sides of the equation
Subtracting a matrix from itself produces a zero matrix, which is an additive identity. This means that the left-hand side of the equation above is equal to .
The right-hand side of this equation contains matrix addition. Recall that we can add matrices of the same order by adding their corresponding entries. We can see that both matrices on the right-hand side of the equation are matrices, so we can compute the matrix addition to obtain
This gives us
In the previous examples, we saw how to combine matrix transposition with other matrix operations, such as addition, subtraction, and scalar multiplication. We can recall a few properties of matrix transposition that are helpful in combining these operations.
Property: Distributive Property of Matrix Transposition
Let and be matrices of the same order. Then,
We know that multiplication of real numbers is distributive, which means
Hence, the matrix transpose behaves like multiplication in that it satisfies the distributive property.
We can also combine matrix transposition with itself. This means that we can apply matrix transposition twice to a matrix. We recall the following result concerning successive applications of matrix transposition.
Property: Successive Applications of Matrix Transposition
Let be a matrix. Then,
This statement tells us that successive applications of matrix transposition bring back the original matrix.
In the next example, we will apply the properties of matrix transposition to compute the given expression.
Example 6: Operations on Matrices Involving the Transpose of a Matrix
Given that determine the value of .
Answer
In this example, we need to compute the expression . We recall that the notation indicates the entry in the row and column of matrix . Hence, we need to find the entries and and sum them. To do this, we need to first compute matrix .
Matrix is given by the expression . We can simplify this expression by applying the properties of matrix transposition. First, recall the distributive property of matrix transposition. Given matrices and of the same order, we have
Applying this property with and , we obtain
We now recall another property of matrix transposition. For any matrix ,
This means that , which leads to
Hence, this leads to
To find matrix , we need to first find the transpose and then subtract from this matrix. Recall that we can find the transpose of a matrix by swapping its rows and columns. This gives us
Then,
Finally, we need to find the matrix subtraction. Recall that we can subtract matrices of the same order by subtracting the corresponding entries. We can see that both matrices on the right-hand side of the equation are matrices, so the matrix subtraction is well defined. Hence, we obtain
Therefore, we have
is the entry of this matrix in the first row and second column, and is the entry in the second row and first column. From the matrix above, we obtain
Hence,
In our final example, we will combine matrix addition and transposition.
Example 7: Combining Matrix Addition and Transposition
Complete the following: If and are two matrices with the same size, then .
Answer
We will solve this example using two different methods. In the first method, we will find the answer by working out the given matrix operations entry by entry to identify what the result is equal to. In the second method, we will apply the properties of matrix transposition to simplify the given expression.
Method 1
We know that adding a superscript indicates transposing the matrix, so the given expression combines matrix addition and transposition. Recall that a combination of matrix operations involving matrix addition, subtraction, scalar multiplication, and transposition is evaluated in the following order:
- Compute the operation within parentheses.
- Compute the transposition.
- Compute the scalar multiplication.
- Compute the matrix addition.
- Compute the matrix subtraction.
In this example, the expression is inside the parentheses, so we need to find this expression first. This expression is also a combination of transposition and addition. Hence, we need to
- transpose matrices and to find and ,
- add the transposed matrices to find ,
- transpose the resulting sum to find .
Let us begin by recalling the notation for a matrix that is useful for defining matrix operations. If we denote for integers and to be a sequence of numbers, then denotes the matrix whose entry in the row and column is given by .
Since matrices and have the same size, we will say that they are both matrices. Hence, we denote
Let us find the transpose of these matrices. We know that the transpose of an matrix is an matrix obtained by swapping its rows and columns. This means
Next, we need to find the sum . Recall that we can add a pair of matrices of the same order by adding each corresponding entry in the matrices. We can see that both and are matrices, so the addition is well defined and is given by
Finally, we can transpose this matrix by swapping its rows with its columns:
Since we know , we can find by swapping and , which leads to
This means
But we also know that the matrix on the right-hand side of this equation can be obtained by adding and . In other words,
This means
Method 2
Recall the distributive property of matrix transposition. Given any two matrices and of the same order,
In this example, and . We know that matrices and are of the same order, say . As noted in the previous method, the transposed matrices and are both of order , which means that they have the same order. Hence,
Next, we recall that the successive application of matrix transposition results in the original matrix. This means
Substituting these expressions into the equation above, we can write
Let us finish by recapping a few important concepts from the explainer.
Key Points
- Given matrices and and a scalar , we define matrix addition, subtraction, scalar multiplication, and transposition as follows:
- A combination of matrix operations involving matrix addition, subtraction, scalar multiplication, and transposition is evaluated in
the following order:
- Compute the operation within parentheses.
- Compute the transposition.
- Compute the scalar multiplication.
- Compute the matrix addition.
- Compute the matrix subtraction.
- For any matrices and of the same order, the useful properties of matrix transposition are
