Lesson Explainer: Adding and Subtracting Matrices Mathematics

In this explainer, we will learn how to add and subtract matrices using the properties of their addition and subtraction.

If ๐ด is an ๐‘›ร—๐‘š matrix, it consists of ๐‘› rows and ๐‘š columns, ๐ด=โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽžโŽŸโŽŸโŽ ,๏Šง๏Šง๏Šง๏Šจ๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏‰๏Š๏Šง๏Š๏Šจ๏Š๏‰ and a square matrix is an ๐‘›ร—๐‘› matrix containing the same number of rows and columns (๐‘š=๐‘›), ๐ด=โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽžโŽŸโŽŸโŽ .๏Šง๏Šง๏Šง๏Šจ๏Šง๏Š๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Š๏Š๏Šง๏Š๏Šจ๏Š๏Š

For this explainer, we will only look at matrices up to and including 3ร—3 matrices (๐‘›,๐‘šโ‰ค3), but the same rules apply for more rows and columns. To add or subtract matrices, you have to add or subtract their corresponding elements.

We can only add or subtract matrices of the same order (i.e., both matrices have the same number of rows and the same number of columns as each other). This makes sense because you cannot perform these operations with the corresponding elements if both matrices do not have the same number of rows and columns.

Adding and Subtracting ๐‘› ร— ๐‘š Matrices:

โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽžโŽŸโŽŸโŽ +โŽ›โŽœโŽœโŽœโŽ๐‘๐‘โ‹ฏ๐‘๐‘๐‘โ‹ฏ๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘๐‘โ‹ฏ๐‘โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽ๐‘Ž+๐‘๐‘Ž+๐‘โ‹ฏ๐‘Ž+๐‘๐‘Ž+๐‘๐‘Ž+๐‘โ‹ฏ๐‘Ž+๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž+๐‘๐‘Ž+๐‘โ‹ฏ๐‘Ž+๐‘โŽžโŽŸโŽŸโŽŸโŽ โŽ›โŽœโŽœโŽ๐‘Ž๐‘Žโ‹ฏ๐‘Ž๐‘Ž๐‘Žโ‹ฏ๐‘Žโ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Ž๐‘Žโ‹ฏ๐‘ŽโŽžโŽŸโŽŸโŽ โˆ’โŽ›โŽœโŽœโŽœโŽ๐‘๐‘โ‹ฏ๐‘๐‘๐‘โ‹ฏ๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘๐‘โ‹ฏ๐‘โŽžโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽ๐‘Žโˆ’๐‘๐‘Žโˆ’๐‘โ‹ฏ๐‘Žโˆ’๐‘๐‘Žโˆ’๐‘๐‘Žโˆ’๐‘โ‹ฏ๐‘Žโˆ’๐‘โ‹ฎโ‹ฎโ‹ฑโ‹ฎ๐‘Žโˆ’๐‘๐‘Žโˆ’๐‘โ‹ฏ๐‘Žโˆ’๐‘โŽžโŽŸโŽŸโŽŸโŽ ๏Šง๏Šง๏Šง๏Šจ๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏‰๏Š๏Šง๏Š๏Šจ๏Š๏‰๏Šง๏Šง๏Šง๏Šจ๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏‰๏Š๏Šง๏Š๏Šจ๏Š๏‰๏Šง๏Šง๏Šง๏Šง๏Šง๏Šจ๏Šง๏Šจ๏Šง๏‰๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏‰๏Šจ๏‰๏Š๏Šง๏Š๏Šง๏Š๏Šจ๏Š๏Šจ๏Š๏‰๏Š๏‰๏Šง๏Šง๏Šง๏Šจ๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏‰๏Š๏Šง๏Š๏Šจ๏Š๏‰๏Šง๏Šง๏Šง๏Šจ๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏‰๏Š๏Šง๏Š๏Šจ๏Š๏‰๏Šง๏Šง๏Šง๏Šง๏Šง๏Šจ๏Šง๏Šจ๏Šง๏‰๏Šง๏‰๏Šจ๏Šง๏Šจ๏Šง๏Šจ๏Šจ๏Šจ๏Šจ๏Šจ๏‰๏Šจ๏‰๏Š๏Šง๏Š๏Šง๏Š๏Šจ๏Š๏Šจ๏Š๏‰๏Š๏‰

To see this in action, consider the 2ร—2 matrices ๐ด=๏€ผ8207๏ˆ,๐ต=๏€ผโˆ’559โˆ’8๏ˆ.

In order to add these matrices, we add corresponding elements: ๐ด+๐ต=๏€ผ8207๏ˆ+๏€ผโˆ’559โˆ’8๏ˆ=๏€พ8+(โˆ’5)2+50+97+(โˆ’8)๏Š=๏€ผ379โˆ’1๏ˆ.

In order to subtract matrix ๐ต from matrix ๐ด, we subtract each element in matrix ๐ต from its corresponding element in matrix ๐ด: ๐ดโˆ’๐ต=๏€ผ8207๏ˆโˆ’๏€ผโˆ’559โˆ’8๏ˆ=๏€พ8โˆ’(โˆ’5)2โˆ’50โˆ’97โˆ’(โˆ’8)๏Š=๏€ผ13โˆ’3โˆ’915๏ˆ.

Now, consider the 3ร—2 matrices ๐ด=๏€673โˆ’25โˆ’1๏Œ,๐ต=๏€2380โˆ’49๏Œ.

Again, letโ€™s add and subtract these matrices by completing each operation on their corresponding elements: ๐ด+๐ต=๏€673โˆ’25โˆ’1๏Œ+๏€2380โˆ’49๏Œ=๏‚6+27+33+8โˆ’2+05+(โˆ’4)โˆ’1+9๏Ž=๏€81011โˆ’218๏Œ,๐ดโˆ’๐ต=๏€673โˆ’25โˆ’1๏Œโˆ’๏€2380โˆ’49๏Œ=๏6โˆ’27โˆ’33โˆ’8โˆ’2โˆ’05โˆ’(โˆ’4)โˆ’1โˆ’9๏=๏€44โˆ’5โˆ’29โˆ’10๏Œ.

Finally, consider the 3ร—3 matrices ๐ด=๏€5622โˆ’334โˆ’1โˆ’5๏Œ,๐ต=๏€1207โˆ’19โˆ’58โˆ’1๏Œ.

Letโ€™s repeat the same operations on these matrices by completing each operation on their corresponding elements as before: ๐ด+๐ต=๏€5622โˆ’334โˆ’1โˆ’5๏Œ+๏€1207โˆ’19โˆ’58โˆ’1๏Œ=๏ƒ5+16+22+02+7โˆ’3+(โˆ’1)3+94+(โˆ’5)โˆ’1+8โˆ’5+(โˆ’1)๏=๏€6829โˆ’412โˆ’17โˆ’6๏Œ,๐ดโˆ’๐ต=๏€5622โˆ’334โˆ’1โˆ’5๏Œโˆ’๏€1207โˆ’19โˆ’58โˆ’1๏Œ=๏‚5โˆ’16โˆ’22โˆ’02โˆ’7โˆ’3โˆ’(โˆ’1)3โˆ’94โˆ’(โˆ’5)โˆ’1โˆ’8โˆ’5โˆ’(โˆ’1)๏Ž=๏€442โˆ’5โˆ’2โˆ’69โˆ’9โˆ’4๏Œ.

There are also properties of addition of matrices that we should be aware of:

  1. Matrix addition is commutative, which means that if ๐ด and ๐ต are two matrices of the same order such that ๐ด+๐ต is defined, then ๐ด+๐ต=๐ต+๐ด, meaning the order in which you perform the addition does not matter. This arises due to the commutative law of addition of numbers, since we perform the addition on the corresponding elements of each matrix.
  2. Matrix addition is associative, which means if ๐ด, ๐ต, and ๐ถ are three matrices of the same order such that ๐ต+๐ถ, ๐ด+(๐ต+๐ถ) and ๐ด+๐ต, (๐ด+๐ต)+๐ถ are defined, then ๐ด+(๐ต+๐ถ)=(๐ด+๐ต)+๐ถ, meaning the sum of the three matrices remains the same regardless of how they are grouped. You can find ๐ด+๐ต first then add the result to ๐ถ or add ๐ด to ๐ต+๐ถ and get the same result.
  3. Matrix addition satisfies the additive identity, which means that if ๐ด is a matrix, then ๐ด+๐‘‚=๐‘‚+๐ด=๐ด, where ๐‘‚ is the zero (or null) matrix of the same order of ๐ด consisting of zeros in each element. The zero matrix ๐‘‚ is called the additive identity for matrices.
  4. Matrix addition satisfies the additive inverse of a matrix, which means that if ๐ด is a matrix, then ๐ด+(โˆ’๐ด)=(โˆ’๐ด)+๐ด=๐‘‚, where ๐‘‚ is the additive identity and โˆ’๐ด is called the additive inverse of ๐ด.

Now, letโ€™s look at a few examples in order to practice and deepen our understanding, starting with the addition of two 1ร—3 matrices.

Example 1: Addition of One-by-Three Matrices

Find (12โˆ’1)+(282).

Answer

In this example, we have to perform the addition of two 1ร—3 matrices.

In order to add these matrices, we add the corresponding elements of each matrix: (12โˆ’1)+(282)=(1+22+8โˆ’1+2)=(3101).

The same method of carrying out operations on all corresponding elements of the two matrices of the same order applies to addition of larger matrices, as we can see in our next example.

Example 2: Addition of Two-by-Two Matrices

Evaluate ๏€ผ811โˆ’37๏ˆ+๏€ผ10โˆ’131๏ˆ.

Answer

In this example, we have to perform the addition of two 2ร—2 matrices.

In order to add these matrices, we add the corresponding elements of each matrix: ๏€ผ811โˆ’37๏ˆ+๏€ผ10โˆ’131๏ˆ=๏€ฝ8+1011+(โˆ’1)โˆ’3+37+1๏‰=๏€ผ181008๏ˆ.

In our next example we subtract one matrix from another.

Example 3: Finding the Difference between Two Given Matrices

Find ๏€ผ79โˆ’50๏ˆโˆ’๏€ผ8โˆ’520๏ˆ.

Answer

In this example, we have to find the difference between 2ร—2 matrices.

In order to subtract these matrices, we subtract the corresponding elements of each matrix: ๏€ผ79โˆ’50๏ˆโˆ’๏€ผ8โˆ’520๏ˆ=๏€ฝ7โˆ’89โˆ’(โˆ’5)โˆ’5โˆ’20โˆ’0๏‰=๏€ผโˆ’114โˆ’70๏ˆ.

Next, let us consider an example that requires us to rearrange and solve a matrix equation using our knowledge of the properties of matrix addition and subtraction.

Example 4: Addition and Subtraction of Matrices

Consider the matrix ๐ด=๏€ผโˆ’1โˆ’1513โˆ’10๏ˆ.

Suppose the sum of matrices ๐ด and ๐ต is ๐ด+๐ต=๏€ผ10โˆ’1012๏ˆ.

Find the matrix ๐ต.

Answer

In this example, we have to find the unknown matrix ๐ต by applying operations on 2ร—3 matrices. Since we can only add or subtract matrices of the same order as we apply the operations on the corresponding elements and both the given matrix and ๐ด are 2ร—3 matrices, ๐ต must also be a 2ร—3 matrix.

In order to find ๐ต, we rearrange the equation to make ๐ต the subject and subtract the corresponding elements of each matrix:๐ด+๐ต=๏€ผ10โˆ’1012๏ˆ.

Since matrix addition is commutative, we know that ๐ด+๐ต=๐ต+๐ด. We can then subtract matrix ๐ด from each side of our equation, leaving ๐ต=๏€ผ10โˆ’1012๏ˆโˆ’๐ด.

Therefore, after substituting ๐ด, we have ๐ต=๏€ผ10โˆ’1012๏ˆโˆ’๏€ผโˆ’1โˆ’1513โˆ’10๏ˆ=๏€พ1โˆ’(โˆ’1)0โˆ’(โˆ’1)โˆ’1โˆ’50โˆ’131โˆ’(โˆ’1)2โˆ’0๏Š=๏€ผ21โˆ’6โˆ’1322๏ˆ.

Now let us use our knowledge of matrix addition and subtraction in an example that requires us to perform an inverse operation to find the answer.

Example 5: Finding an Unknown Matrix by Applying Operations on Matrices Involving the Zero Matrix

Given that ๐‘‹+๏€ผโˆ’6โˆ’865๏ˆ=๐‘‚, where ๐‘‚ is the 2ร—2 zero matrix, find the value of ๐‘‹.

Answer

In this example, we have to find the unknown matrix ๐‘‹ by applying operations on 2ร—2 matrices. Since we can only add or subtract matrices of the same order as we apply the operations on the corresponding elements and both the given matrix and the zero matrix are 2ร—2 matrices, ๐‘‹ must also be a 2ร—2 matrix.

In order to find the unknown matrix ๐‘‹, we will rearrange the equation to make ๐‘‹ the subject and then we will subtract the corresponding elements of the given matrix from the zero matrix: ๐‘‹+๏€ผโˆ’6โˆ’865๏ˆ=๐‘‚.

Since matrix addition is commutative, we can subtract the given matrix from each side of our equation. As the first matrix is the zero matrix, this has the effect of switching signs on each element of the second matrix: ๐‘‹=๐‘‚โˆ’๏€ผโˆ’6โˆ’865๏ˆ=๏€ผ0000๏ˆโˆ’๏€ผโˆ’6โˆ’865๏ˆ=๏€ฝ0โˆ’(โˆ’6)0โˆ’(โˆ’8)0โˆ’60โˆ’5๏‰=๏€ผ68โˆ’6โˆ’5๏ˆ.

In our final example, we will use our knowledge of the commutativity of matrix addition to help us perform a calculation involving the addition and subtraction of three matrices.

Example 6: Commutative and Associative Properties of Operations on Matrices

Given that ๐ด=๏€ผ43โˆ’13๏ˆ,๐ต=๏€ผโˆ’1023๏ˆ,๐ถ=๏€ผโˆ’5107๏ˆ, find ๐ด+๐ตโˆ’๐ถ.

Answer

In this example, we have to perform addition and subtraction operations on three 2ร—2 matrices.

In order to do this, we perform each operation on the corresponding elements of each matrix: ๐ด+๐ตโˆ’๐ถ=๏€ผ43โˆ’13๏ˆ+๏€ผโˆ’1023๏ˆโˆ’๏€ผโˆ’5107๏ˆ=๏€ฝ4+(โˆ’1)โˆ’(โˆ’5)3+0โˆ’1โˆ’1+2โˆ’03+3โˆ’7๏‰=๏€ผ821โˆ’1๏ˆ.

Matrix addition is commutative, so if we consider subtracting a matrix as being equivalent to adding the negative of the matrix, we can see that we could perform the calculation in several different ways and still get the same answer:

Matrix addition is also associative, so we can group the items in different ways and still get the same answer:

Key Points

  • To add two matrices, we need to add each corresponding element.
  • To subtract one matrix from another, we need to subtract the corresponding elements.
  • To perform these operations, the order of each matrix must be the same.
  • Matrix addition is commutative and associative.
  • Matrix addition satisfies the additive identity and additive inverse properties.

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