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Lesson 3 of 8

Matrices · Lesson 3 of 8

Addition, Scaling, and Matrix Equations

“Combine comparable data, scale every entry, and use matrix laws to solve equations.”

Learning Objectives

• Add and subtract matrices of the same order and explain the order restriction. • Multiply a matrix by a scalar and interpret negatives and percentage changes. • Justify the addition laws and scalar distributive laws entry by entry. • Solve additive equations for unknown matrices and unknown entries. • Interpret combined sales, differences, and profit matrices with consistent labels.

Combining matching quantities

Matrix addition describes combining like information. If two factories record the same categories in the same positions, we can add corresponding entries to obtain combined production. The mathematical rule is simple, but its interpretation depends on matching categories as well as matching shapes.

Price categoryFactory A: boysFactory A: girlsFactory B: boysFactory B: girls
180609050
275657055
390857575

Each factory supplies a 3 × 2 matrix: rows are price categories and columns are boys’ and girls’ sport shoes. For category 1, the combined boys’ quantity is 80+90 = 170 and the combined girls’ quantity is 60+50 = 110. Repeating the same operation at every matching position gives the total production matrix.

Entrywise additionLaTeX
Both inputs and the sum have order m × n. The (i,j) entry of the sum combines the (i,j) entries of the two inputs.
Definition
Sum of matrices

For matrices A and B of the same order, A+B is obtained by adding each entry of A to the corresponding entry of B. If the orders differ, their sum is not defined.

Why insist on the same order? A 2 × 2 matrix has no third-column position to pair with the third column of a 2 × 3 matrix. There is no instruction to “add the overlapping part and ignore the rest”. On the set of all matrices of one fixed order, however, addition always gives another matrix of that order. In that sense it is a binary operation on that set.

Example — Adding entries with different number types

Problem
Let A = [√3,1,−1;2,3,0] and B = [2,√5,1;−2,3,1/2]. Find A+B.

  1. 1.Both matrices have order 2 × 3, so their sum is defined. Pair entries at identical addresses.
  2. 2.The first row of the sum is [√3+2, 1+√5, −1+1] = [2+√3, 1+√5, 0].
  3. 3.The second row is [2+(−2), 3+3, 0+1/2] = [0,6,1/2].
  4. 4.Thus A+B = [2+√3, 1+√5, 0; 0,6,1/2]. The rule works for irrational numbers and fractions just as it does for integers.

Scaling, negatives, and differences

Multiplying a matrix by one ordinary number applies the same scale change to every entry. This is useful for doubling quantities, converting units, or taking a percentage of each category. It also gives a precise meaning to the negative of a matrix and hence to subtraction.

Definition
Scalar multiplication

For a scalar k and matrix A = [aᵢⱼ], the matrix kA has entries kaᵢⱼ. Its order is the same as A’s; every entry is multiplied by k.

Scaling and subtractionLaTeX
The numbers k and −1 act on every entry. Subtraction requires the same order for A and B, just as addition does.

Doubling factory A’s production gives 2A = [160,120;150,130;180,170]. Multiplying by 0 gives the zero matrix of the same order. Multiplying by −1 changes every entry’s sign; it does not rearrange the rows or columns. To find A−B, we may either subtract matching entries directly or add A to the entrywise negative of B.

Example — Combining scaling and subtraction

Problem
For A = [1,2,3;2,3,1] and B = [3,−1,3;−1,0,2], calculate 2A−B.

  1. 1.First double each entry of A: 2A = [2,4,6;4,6,2].
  2. 2.Subtract B entry by entry. The first row becomes [2−3, 4−(−1), 6−3] = [−1,5,3].
  3. 3.The second row becomes [4−(−1), 6−0, 2−2] = [5,6,0].
  4. 4.Therefore 2A−B = [−1,5,3;5,6,0]. Subtracting a negative entry adds its positive value; the sign belongs to the entry being subtracted.
A scalar acts on every entry

In 2A, multiplying only the diagonal or only the first row changes the operation. Also distinguish −A from A−I: the first negates every entry, while the second is defined only for square A of matching order and subtracts 1 only on the diagonal.

Why the addition laws work

Because addition and scaling act separately at each address, their laws come from the corresponding laws for real numbers. We can justify an entire matrix equality by checking a general (i,j) entry. This avoids assuming that every familiar numerical rule will automatically hold for every matrix operation.

Addition lawsLaTeX
A, B, C, and O have one common order. These are commutativity, associativity, additive identity, and additive inverse respectively.

For commutativity, the (i,j) entries of A+B and B+A are aᵢⱼ+bᵢⱼ and bᵢⱼ+aᵢⱼ. They are equal because real-number addition is commutative. For associativity, the relevant entries are (aᵢⱼ+bᵢⱼ)+cᵢⱼ and aᵢⱼ+(bᵢⱼ+cᵢⱼ), equal by associativity of real-number addition. Since these equalities hold at every address, the corresponding matrices are equal.

Adding the matching zero matrix leaves each entry unchanged, so O is the additive identity. Adding −A makes each entry aᵢⱼ+(−aᵢⱼ) = 0, so −A is the additive inverse of A. The word “inverse” here refers to addition. Later, an inverse matrix will refer to multiplication and will have a different defining equation.

Scalar distributive lawsLaTeX
A and B have the same order, while k and ℓ are real numbers. The first law distributes over a matrix sum; the second distributes over a scalar sum.

The (i,j) entry of k(A+B) is k(aᵢⱼ+bᵢⱼ) = kaᵢⱼ+kbᵢⱼ, exactly the corresponding entry of kA+kB. Similarly, (k+ℓ)aᵢⱼ = kaᵢⱼ+ℓaᵢⱼ establishes the second law. Consecutive scalings also behave as expected: k(ℓA) has entries k(ℓaᵢⱼ) = (kℓ)aᵢⱼ. These arguments all preserve the original addresses.

Solving additive matrix equations

An equation involving unknown matrices can be manipulated using the laws just established. Adding the same matrix to both sides preserves equality, and multiplying both sides by a nonzero scalar allows us to isolate an unknown. After that, equality of corresponding entries completes the calculation.

Example — Isolating an unknown matrix

Problem
Given A = [8,0;4,−2;3,6] and B = [2,−2;4,2;−5,1], solve 2A+3X = 5B.

  1. 1.All terms must have order 3 × 2. Add −2A to both sides and use associativity, commutativity, and the additive inverse to obtain 3X = 5B−2A.
  2. 2.Multiply by the scalar 1/3: X = (1/3)(5B−2A). This is division by the number 3, not division by a matrix.
  3. 3.Calculate 5B = [10,−10;20,10;−25,5] and 2A = [16,0;8,−4;6,12]. Their difference is [−6,−10;12,14;−31,−7].
  4. 4.Scale every entry by 1/3 to get X = [−2,−10/3;4,14/3;−31/3,−7/3].
  5. 5.Checking 2A+3X recovers [10,−10;20,10;−25,5], which equals 5B. The check uses the full arrays, including fractional entries.
Example — Recovering two matrices from their sum and difference

Problem
Find X and Y if X+Y = [5,2;0,9] and X−Y = [3,6;0,−1].

  1. 1.Add the equations. On the left, Y and −Y cancel; on the right, add matching entries. Thus 2X = [8,8;0,8].
  2. 2.Multiply by 1/2 to obtain X = [4,4;0,4].
  3. 3.Subtract the second equation from the first. The left side is X+Y−(X−Y) = 2Y, so 2Y = [2,−4;0,10].
  4. 4.Therefore Y = [1,−2;0,5]. Their sum is [5,2;0,9] and their difference is [3,6;0,−1], checking both conditions.
  5. 5.The useful general pattern is X = (S+D)/2 and Y = (S−D)/2 when X+Y = S and X−Y = D. All four matrices have one common order.
Example — Combining operations with entry equality

Problem
Solve 2[x,5;7,y−3] + [3,−4;1,2] = [7,6;15,14].

  1. 1.First scale and add to obtain [2x+3,6;15,2y−4] = [7,6;15,14].
  2. 2.The top-left entries give 2x+3 = 7, so x = 2. The bottom-right entries give 2y−4 = 14, so y = 9.
  3. 3.The other two entries are 6 = 6 and 15 = 15. They confirm that these positions impose no contradictory conditions.
  4. 4.Substitute x = 2 and y = 9 into the original equation: 2[2,5;7,6]+[3,−4;1,2] = [7,6;15,14].

Reading the results in context

A calculated matrix is useful only when we interpret its entries in the original categories. The same pair of data matrices can support totals, changes, and scaled returns. These are three different questions, so their operations and units must be kept distinct.

FarmerSeptember: BasmatiSeptember: PermalSeptember: NauraOctober: BasmatiOctober: PermalOctober: Naura
Ramkishan1000020000300005000100006000
Gurcharan Singh500003000010000200001000010000
Example — Combined sales, decreases, and profit

Problem
The table records sales in rupees. Let A record September and B record October, with farmers as rows and rice varieties as columns. Find the combined sales, the decrease from September to October, and the stipulated profit of 2% of October sales in each category.

  1. 1.Write A = [10000,20000,30000;50000,30000,10000] and B = [5000,10000,6000;20000,10000,10000]. Their identical labels make corresponding entries comparable.
  2. 2.Combined sales are A+B = [15000,30000,36000;70000,40000,20000] rupees. For instance, the first entry combines the same farmer and variety over two months.
  3. 3.The decrease is September minus October: A−B = [5000,10000,24000;30000,20000,0] rupees. The final zero means unchanged Naura sales for Gurcharan Singh.
  4. 4.Two percent means multiplication by 2/100 = 0.02. The profit matrix is 0.02B = [100,200,120;400,200,200] rupees.
  5. 5.Each profit entry refers to one farmer and one variety. If a total profit per farmer is also wanted, sum across that farmer’s row: ₹420 and ₹800.
  6. 6.The phrase “decrease” fixes the subtraction order. If October sales had exceeded September sales in a category, the corresponding decrease entry would be negative, indicating an increase.

Quiz

Quick check

When is A+B defined?

Quick check

For A = [1,−2;0,3], what is −2A?

Quick check

Which matrix is the additive identity for matrices of order 2 × 3?

Quick check

If X+Y = S and X−Y = D, which expression gives Y?

Quick check

For same-order A and B, which equality always holds?

Quick check

A records last month’s sales and B this month’s. An entry of A−B is −600. What does it mean?

Practice Problems

Practice Problems
  1. For A = [2,4;3,2], B = [1,3;−2,5], C = [−2,5;3,4], calculate A+B, A−B, and 3A−C. Keep the positions fixed throughout.
  2. Compute these sums: (a) [a,b;−b,a]+[a,b;b,a]; (b) [a²+b²,b²+c²;a²+c²,a²+b²]+[2ab,2bc;−2ac,−2ab]; (c) [−1,4,−6;8,5,16;2,8,5]+[12,7,6;8,0,5;3,2,4]; (d) [cos²x,sin²x;sin²x,cos²x]+[sin²x,cos²x;cos²x,sin²x].
  3. Use A = [1,2,−3;5,0,2;1,−1,1], B = [3,−1,2;4,2,5;2,0,3], C = [4,1,2;0,3,2;1,−2,3]. Calculate A+B and B−C, then verify A+(B−C) = (A+B)−C.
  4. Calculate 3A−5B for A = [2/3,1,5/3;1/3,2/3,4/3;7/3,2,2/3] and B = [2/5,3/5,1;1/5,2/5,4/5;7/5,6/5,2/5]. Explain the cancellation pattern.
  5. Simplify cos θ[cos θ,sin θ;−sin θ,cos θ] + sin θ[sin θ,−cos θ;cos θ,sin θ]. Use trigonometric identities at corresponding entries.
  6. Find X and Y: (a) X+Y = [7,0;2,5] and X−Y = [3,0;0,3]; (b) 2X+3Y = [2,3;4,0] and 3X+2Y = [2,−2;−1,5]. Show the elimination steps.
  7. Find X when Y = [3,2;1,4] and 2X+Y = [1,0;−3,2]. Verify by substitution.
  8. Find x and y in 2[1,3;0,x]+[y,0;1,2] = [5,6;1,8].
  9. Solve for x, y, z, t in 2[x,z;y,t]+3[1,−1;0,2] = 3[3,5;4,6].
  10. Find x and y from x[2;3]+y[−1;1] = [10;5]. Translate the column equality into two scalar equations.
  11. Solve 3[x,y;z,w] = [x,6;−1,2w]+[4,x+y;z+w,3] for x, y, z, w. Check the entries after solving.
  12. X and Z have orders 2 × n and 2 × p. If n = p, find the order of 7X−5Z and explain why it is defined.
  13. Prove entry by entry that A−B = −(B−A). Explain why this does not make subtraction commutative.

Key Takeaways

Key Takeaways

• Addition and subtraction pair corresponding entries and require identical orders. • Scalar multiplication changes every entry by the same factor and preserves order. • O is the additive identity and −A is the additive inverse of A. • Addition is commutative and associative; scalar multiplication distributes over matrix sums and scalar sums. • Additive matrix equations can be solved by valid operations on both sides, followed by entrywise checks. • In applications, retain labels and units and choose subtraction order from the question.

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Matrix Types and Equality

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How Matrix Multiplication Works