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Quizzes > High School Quizzes > Mathematics

Multiply 2x2 Matrix by 1x2 Practice Quiz

Sharpen your matrix multiplication skills through practice

Difficulty: Moderate
Grade: Grade 10
Study OutcomesCheat Sheet
Colorful paper art promoting Matrix Multiply Magic, a high school math quiz.

Is the multiplication of a 2x2 matrix by a 1x2 matrix defined using standard matrix multiplication rules?
No, because the inner dimensions do not match.
Yes, it results in a 2x1 matrix.
Yes, it results in a 1x2 matrix.
Yes, it results in a 2x2 matrix.
Matrix multiplication requires that the number of columns in the first matrix equals the number of rows in the second. In this case, the 2x2 matrix has 2 columns and the 1x2 matrix has 1 row, so the multiplication is not defined.
What is the necessary condition for two matrices A (of size m×n) and B (of size p×q) to be multiplicatively compatible?
Both matrices must be square.
The number of columns of A must equal the number of rows of B.
The number of rows of A must equal the number of columns of B.
Both matrices must have the same dimensions.
For the product AB to be defined, the number of columns in A must equal the number of rows in B, regardless of whether the matrices are square or have the same overall dimensions.
What is the result of multiplying the row vector [5, 6] by the matrix [[1, 2], [3, 4]]?
[18, 32]
[23, 34]
[19, 28]
[26, 38]
To compute the product, take the dot product of the row vector with each column of the matrix. The first element is 5×1 + 6×3 = 23 and the second is 5×2 + 6×4 = 34.
Which operations are combined when performing matrix multiplication?
Both multiplication and addition
Only multiplication
Division and multiplication
Only addition
Matrix multiplication involves multiplying corresponding elements and then summing the resulting products. This combination of multiplication and addition is fundamental to the matrix product operation.
If A = [[a, b], [c, d]] and x is a column vector [p, q], what is the expression for the first element of the product Ax?
ap - bq
ap + bq
a*p * b*q
a + p + b + q
The first element of the product Ax is found by taking the dot product of the first row of A and the column vector x, which yields ap + bq.
Given A = [[2, 3], [4, 5]] and B as the 2x2 identity matrix, what is the product A × B?
[[8, 15], [8, 15]]
[[2, 0], [4, 0]]
[[2, 3], [4, 5]]
[[1, 0], [0, 1]]
Multiplying any matrix by the identity matrix returns the original matrix. Since B is the identity matrix, A × B equals A.
Which property does matrix multiplication always satisfy?
Associative property
Both commutative and associative properties
Commutative property
Neither commutative nor associative properties
Matrix multiplication is associative, meaning (AB)C = A(BC), but it is generally not commutative. Changing the order can lead to different results or may even be undefined.
Calculate the product of the row vector [3, -1] and the matrix [[2, 4], [5, 0]].
[1, 12]
[1, -12]
[-1, 12]
[8, 12]
Multiply the row vector by each column: 3×2 + (-1)×5 gives 1, and 3×4 + (-1)×0 gives 12. This yields the product [1, 12].
For matrices A (m×n) and B (p×q), which of the following statements is true about the product AB?
The product is defined if n equals q.
The product is defined if m equals p.
The product is defined if m equals q.
The product is defined if n equals p.
For the product AB to be defined, the number of columns in A (n) must equal the number of rows in B (p). This condition ensures that each element can be computed properly.
Given A = [[1, 2], [3, 4]] and B = [[0, 1], [1, 0]], compute the product AB.
[[1, 2], [3, 4]]
[[2, 1], [4, 3]]
[[0, 1], [1, 0]]
[[3, 4], [1, 2]]
The product AB is calculated by taking dot products of rows of A with columns of B. This yields the entry 1×0 + 2×1 = 2 for the first element and similarly for the others, resulting in [[2, 1], [4, 3]].
Which statement best describes the effect of changing the order of factors in matrix multiplication?
The order of factors has no impact on the result.
Matrix multiplication is only defined when the matrices are identical.
In general, AB does not equal BA.
AB is always equal to BA.
Matrix multiplication is generally not commutative; therefore, swapping the order of the matrices typically changes the result or may lead to an undefined product.
Compute the product of A = [[-1, 3], [2, 4]] and B = [[2, 0], [1, -5]].
[[1, -15], [8, -20]]
[[-1, 15], [-8, 20]]
[[1, -5], [4, -10]]
[[2, -15], [8, -25]]
Each entry in the product is the dot product of a row from A and a column from B. For example, the top left entry is (-1×2) + (3×1) = 1. The remaining entries are computed similarly.
If a 1x2 matrix is multiplied by a 2x2 matrix, what are the dimensions of the resulting product?
1x2
2x2
2x1
1x1
The product of a 1x2 matrix and a 2x2 matrix has the number of rows of the first matrix and the number of columns of the second matrix, which results in a 1x2 matrix.
Evaluate the product: [7, -3] multiplied by the matrix [[0, 2], [5, 1]].
[15, -11]
[15, 11]
[-15, -11]
[-15, 11]
Compute the dot product of the row vector with each column of the matrix: 7×0 + (-3)×5 = -15 and 7×2 + (-3)×1 = 11, resulting in the product [-15, 11].
Which of the following products is NOT defined?
A 1x3 matrix multiplied by a 3x1 matrix
A 3x2 matrix multiplied by a 2x4 matrix
A 2x2 matrix multiplied by a 1x2 matrix
A 4x3 matrix multiplied by a 3x3 matrix
For matrix multiplication to be defined, the number of columns in the first matrix must equal the number of rows in the second. In the case of a 2x2 matrix and a 1x2 matrix, 2 does not equal 1, so the product is not defined.
Express the product of the 1x2 row vector [x, y] and the 2x2 matrix [[a, b], [c, d]] in algebraic terms.
[ax - cy, bx - dy]
[ax + by, cx + dy]
[ax + cy, bx + dy]
[a + x, c + y]
When multiplying a 1x2 row vector by a 2x2 matrix, the product is determined by taking the dot product of the row with each column. The first entry is x×a + y×c and the second is x×b + y×d, which corresponds to option A.
Solve for x and y given that [x, y] multiplied by the matrix [[2, -1], [4, 3]] equals [10, 5].
[5, 0]
[1, 2]
No solution
[2, 1]
Setting up the equations from the multiplication gives: 2x + 4y = 10 and (-1)x + 3y = 5. Solving these equations yields x = 1 and y = 2.
If A is an invertible 2x2 matrix, what is the result of multiplying A by its inverse?
The zero matrix
The matrix A itself
The identity matrix
The transpose of A
By definition, an invertible matrix A has an inverse A❻¹ such that A × A❻¹ equals the identity matrix. This is a fundamental property of inverse matrices.
Let X and Y be 1x2 matrices and A be a 2x2 matrix. Which of the following demonstrates the distributive property for matrix multiplication?
A × (X + Y) = AX + AY
(X + Y) × A = XA + YA
X × (A + Y) = XA + XY
(A + X) × Y = AY + XY
Since X and Y are 1x2 matrices, their sum is also 1x2 and can be multiplied on the right by the 2x2 matrix A. This shows that (X + Y) × A is equal to XA + YA, illustrating the distributive property.
If [p, q] multiplied by the matrix [[3, 2], [1, 4]] yields [11, 8], find the values of p and q.
[1/5, 18/5]
[18/5, 1/5]
[8, 11]
[11, 8]
Multiplying [p, q] by the given matrix results in the equations 3p + q = 11 and 2p + 4q = 8. Solving these equations gives p = 18/5 and q = 1/5, which is the correct answer.
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Study Outcomes

  1. Understand the concept of matrix dimensions and compatibility in multiplication.
  2. Apply matrix multiplication rules to multiply a 2x2 matrix by a 1x2 matrix.
  3. Calculate the entries of the resulting matrix accurately.
  4. Analyze and identify common errors during the multiplication process.
  5. Verify the accuracy of the computed matrix product.

2x2 Matrix by 1x2 Cheat Sheet

  1. Understand matrix dimensions - Get cozy with the sizes: multiplying a 2×2 matrix by a 2×1 matrix always gives you a 2×1 result because the "inner" dimensions match and the "outer" dimensions define the shape. It's like fitting puzzle pieces together! socratic.org
  2. Ensure compatibility - Before you start crunching numbers, check that the number of columns in the first matrix equals the number of rows in the second. If they don't match, no multiplication magic can happen. RMIT Learning Lab
  3. Follow the multiplication process - Multiply each element in a row of the first matrix by the corresponding element in a column of the second, then sum those products. Repeat for every row - column pair, and voilà, you've built the whole result matrix. GeeksforGeeks
  4. Practice with examples - The more you try, the more you fly! Working through varied sample problems helps cement the steps in your brain. Always challenge yourself with both simple and tricky cases. RMIT Learning Lab
  5. Remember the order matters - Unlike regular numbers, matrices aren't commutative: A×B doesn't always equal B×A. Swapping them can completely change the outcome, so follow your specified order. RMIT Learning Lab
  6. Use the identity matrix - The identity matrix is your multiplication BFF: multiply any matrix by it, and you'll get the original matrix back. Think of it as the "1" of the matrix world. RMIT Learning Lab
  7. Visualize the process - Sometimes a quick video can turn confusion into clarity. Seeing the row-by-column dance in motion makes the rules click faster. Khan Academy Video
  8. Check your work - After you've done the multiplication, double-check your sums and positions. A fresh look or a peer review can catch sneaky sign errors or typos. RMIT Learning Lab
  9. Understand scalar multiplication - Multiplying a matrix by a scalar means scaling every single entry by that number. It's like turning up the volume on each element simultaneously! Math Warehouse
  10. Stay positive and keep practicing - Mastery comes with consistency and a growth mindset. Celebrate small wins, tackle new problems every day, and watch your matrix skills soar! Khan Academy
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