Properties of Matrix Operations - Pre-Calculus
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What is a valid conclusion if $BA=CA$ for square matrices and $A$ is invertible?
What is a valid conclusion if $BA=CA$ for square matrices and $A$ is invertible?
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$B=C$. Multiply both sides by $A^{-1}$ on the right.
$B=C$. Multiply both sides by $A^{-1}$ on the right.
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What is a valid conclusion if $AB=AC$ for square matrices and $A$ is invertible?
What is a valid conclusion if $AB=AC$ for square matrices and $A$ is invertible?
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$B=C$. Multiply both sides by $A^{-1}$ to cancel $A$.
$B=C$. Multiply both sides by $A^{-1}$ to cancel $A$.
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What is the correct simplification of $(AB+CB)D$ using distributive properties?
What is the correct simplification of $(AB+CB)D$ using distributive properties?
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$ABD+CBD$. Factor out $B$ first: $((A+C)B)D = (AB+CB)D = ABD + CBD$.
$ABD+CBD$. Factor out $B$ first: $((A+C)B)D = (AB+CB)D = ABD + CBD$.
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What property states that matrix multiplication is associative?
What property states that matrix multiplication is associative?
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Associative: $(AB)C=A(BC)$ (when defined). Grouping doesn't affect the result when multiplying matrices.
Associative: $(AB)C=A(BC)$ (when defined). Grouping doesn't affect the result when multiplying matrices.
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What property of matrix multiplication usually fails for square matrices: commutative, associative, or distributive?
What property of matrix multiplication usually fails for square matrices: commutative, associative, or distributive?
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Commutative usually fails: in general $AB\ne BA$. Order matters in matrix multiplication, unlike regular numbers.
Commutative usually fails: in general $AB\ne BA$. Order matters in matrix multiplication, unlike regular numbers.
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Identify the incorrect step: from $(A+B)C$ to $AC+CB$. What is the correct result?
Identify the incorrect step: from $(A+B)C$ to $AC+CB$. What is the correct result?
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Correct: $AC+BC$. Order matters: $CB ≠ BC$, so keep original order.
Correct: $AC+BC$. Order matters: $CB ≠ BC$, so keep original order.
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Identify the incorrect step: from $A(B+C)$ to $AB+CA$. What is the correct result?
Identify the incorrect step: from $A(B+C)$ to $AB+CA$. What is the correct result?
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Correct: $AB+AC$. Order matters: $AC ≠ CA$, so keep original order.
Correct: $AB+AC$. Order matters: $AC ≠ CA$, so keep original order.
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Which statement is always true for square matrices of the same size: $AB=BA$ or $(AB)C=A(BC)$?
Which statement is always true for square matrices of the same size: $AB=BA$ or $(AB)C=A(BC)$?
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$(AB)C=A(BC)$. Associativity always holds; commutativity doesn't.
$(AB)C=A(BC)$. Associativity always holds; commutativity doesn't.
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Which expression is guaranteed equal to $A(BC)$ by associativity: $(AB)C$ or $B(AC)$?
Which expression is guaranteed equal to $A(BC)$ by associativity: $(AB)C$ or $B(AC)$?
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$(AB)C$. Associativity allows regrouping without changing order.
$(AB)C$. Associativity allows regrouping without changing order.
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What is the correct expansion of $(A+B)(C+D)$ using distributive properties?
What is the correct expansion of $(A+B)(C+D)$ using distributive properties?
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$AC+AD+BC+BD$. Each term in first parentheses multiplies each in second.
$AC+AD+BC+BD$. Each term in first parentheses multiplies each in second.
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What is the correct simplification of $(A-B)C$ using distributive properties?
What is the correct simplification of $(A-B)C$ using distributive properties?
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$AC-BC$. Apply right distributive property: $(A-B)C = AC - BC$.
$AC-BC$. Apply right distributive property: $(A-B)C = AC - BC$.
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What is the correct simplification of $A(B-C)$ using distributive properties?
What is the correct simplification of $A(B-C)$ using distributive properties?
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$AB-AC$. Apply left distributive property: $A(B-C) = AB - AC$.
$AB-AC$. Apply left distributive property: $A(B-C) = AB - AC$.
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What property states right distributivity of matrix multiplication over addition?
What property states right distributivity of matrix multiplication over addition?
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Right distributive: $(A+B)C=AC+BC$. The sum distributes over matrix $C$ on the right.
Right distributive: $(A+B)C=AC+BC$. The sum distributes over matrix $C$ on the right.
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What property states left distributivity of matrix multiplication over addition?
What property states left distributivity of matrix multiplication over addition?
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Left distributive: $A(B+C)=AB+AC$. Matrix $A$ distributes over the sum inside parentheses.
Left distributive: $A(B+C)=AB+AC$. Matrix $A$ distributes over the sum inside parentheses.
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What is a correct way to reorder factors in a product of matrices: change order or regroup with parentheses?
What is a correct way to reorder factors in a product of matrices: change order or regroup with parentheses?
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Regroup only: you may use $(AB)C=A(BC)$, not reorder factors. Associativity allows regrouping, not reordering.
Regroup only: you may use $(AB)C=A(BC)$, not reorder factors. Associativity allows regrouping, not reordering.
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Which statement is generally false for square matrices: $(AB)^T=B^TA^T$ or $AB=BA$?
Which statement is generally false for square matrices: $(AB)^T=B^TA^T$ or $AB=BA$?
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$AB=BA$ is generally false. Commutativity fails but transpose reversal always works.
$AB=BA$ is generally false. Commutativity fails but transpose reversal always works.
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What is the correct simplification of $A(BC+BD)$ using distributive properties?
What is the correct simplification of $A(BC+BD)$ using distributive properties?
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$ABC+ABD$. Factor out $B$ first: $A(B(C+D)) = AB(C+D) = ABC + ABD$.
$ABC+ABD$. Factor out $B$ first: $A(B(C+D)) = AB(C+D) = ABC + ABD$.
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What is the correct statement about canceling in $AB=CB$ when $B$ may be noninvertible?
What is the correct statement about canceling in $AB=CB$ when $B$ may be noninvertible?
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No conclusion; $AB=CB$ does not imply $A=C$. Without invertibility, can't cancel matrices.
No conclusion; $AB=CB$ does not imply $A=C$. Without invertibility, can't cancel matrices.
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What is the correct statement about canceling $A$ in $AB=AC$ when $A$ may be noninvertible?
What is the correct statement about canceling $A$ in $AB=AC$ when $A$ may be noninvertible?
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No conclusion; $AB=AC$ does not imply $B=C$. Without invertibility, can't cancel matrices.
No conclusion; $AB=AC$ does not imply $B=C$. Without invertibility, can't cancel matrices.
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Identify the correct expansion of $(A+B)(C+D)$ using distributive properties.
Identify the correct expansion of $(A+B)(C+D)$ using distributive properties.
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$AC + AD + BC + BD$. Apply distributive property twice: $(A+B)$ to each term, then expand.
$AC + AD + BC + BD$. Apply distributive property twice: $(A+B)$ to each term, then expand.
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Find and correct the false statement: "If $AB=AC$, then $B=C$ for matrices."
Find and correct the false statement: "If $AB=AC$, then $B=C$ for matrices."
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False; cancellation can fail if $A$ is not invertible. Unlike numbers, matrices can't always be cancelled from equations.
False; cancellation can fail if $A$ is not invertible. Unlike numbers, matrices can't always be cancelled from equations.
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What is the left distributive property of matrix multiplication over addition?
What is the left distributive property of matrix multiplication over addition?
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$A(B+C) = AB + AC$. Multiplying $A$ by a sum equals the sum of individual products.
$A(B+C) = AB + AC$. Multiplying $A$ by a sum equals the sum of individual products.
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What is the right distributive property of matrix multiplication over addition?
What is the right distributive property of matrix multiplication over addition?
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$(A+B)C = AC + BC$. A sum multiplied by $C$ equals the sum of individual products.
$(A+B)C = AC + BC$. A sum multiplied by $C$ equals the sum of individual products.
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What is the additive identity matrix equation for any square matrix $A$?
What is the additive identity matrix equation for any square matrix $A$?
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$A + 0 = A$. The zero matrix acts as the additive identity for matrices.
$A + 0 = A$. The zero matrix acts as the additive identity for matrices.
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What is the multiplicative identity matrix equation for any $n \times n$ matrix $A$?
What is the multiplicative identity matrix equation for any $n \times n$ matrix $A$?
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$AI = IA = A$. The identity matrix $I$ leaves any matrix unchanged when multiplied.
$AI = IA = A$. The identity matrix $I$ leaves any matrix unchanged when multiplied.
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What is the zero-product property for matrices involving the zero matrix $0$?
What is the zero-product property for matrices involving the zero matrix $0$?
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$A^0 = 0$ and $0A = 0$. Multiplying any matrix by the zero matrix yields the zero matrix.
$A^0 = 0$ and $0A = 0$. Multiplying any matrix by the zero matrix yields the zero matrix.
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What is the scalar distributive property with a scalar $k$ and matrices $A$ and $B$?
What is the scalar distributive property with a scalar $k$ and matrices $A$ and $B$?
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$k(A+B) = kA + kB$. Scalars distribute over matrix addition like regular distributive property.
$k(A+B) = kA + kB$. Scalars distribute over matrix addition like regular distributive property.
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Using your results for $AB$ and $BA$, what conclusion follows about commutativity here?
Using your results for $AB$ and $BA$, what conclusion follows about commutativity here?
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$AB \ne BA$, so multiplication is not commutative. Since the products differ, matrix multiplication isn't commutative.
$AB \ne BA$, so multiplication is not commutative. Since the products differ, matrix multiplication isn't commutative.
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What condition on square matrix $A$ makes left cancellation valid: $AB=AC \Rightarrow B=C$?
What condition on square matrix $A$ makes left cancellation valid: $AB=AC \Rightarrow B=C$?
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$A$ must be invertible. Only invertible matrices allow cancellation: multiply both sides by $A^{-1}$.
$A$ must be invertible. Only invertible matrices allow cancellation: multiply both sides by $A^{-1}$.
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Which statement is always true for square matrices: $AB=BA$, $A(B+C)=AB+AC$, or $A(BC)=(AB)C$?
Which statement is always true for square matrices: $AB=BA$, $A(B+C)=AB+AC$, or $A(BC)=(AB)C$?
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$A(B+C)=AB+AC$ and $A(BC)=(AB)C$ are always true. Distributive and associative properties hold, but commutativity doesn't.
$A(B+C)=AB+AC$ and $A(BC)=(AB)C$ are always true. Distributive and associative properties hold, but commutativity doesn't.
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