Matrix by Scalar Calculator

Enter a 3×3 matrix and a scalar k to compute kA, where (kA)ij = k · aij, along with the determinant, trace, and sum of the resulting matrix.

Quick Facts

Scalar multiplication rule
(kA)₀₀ = k · a₀₀
Every entry is multiplied by the same scalar k; the matrix size never changes.
Determinant scaling
det(kA) = kⁿ · det(A)
For an n×n matrix, each of the n rows contributes a factor of k, so a 3×3 matrix scales by k³.
Trace scaling
tr(kA) = k · tr(A)
The trace is the sum of the diagonal entries, so it scales linearly with k.

Your Results

Calculated
Resultant Matrix (kA)
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Each entry multiplied by k
Determinant of kA
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det(kA) = k³·det(A) for a 3×3 matrix
Trace of kA
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Sum of the diagonal entries of kA
Sum of All Entries
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Sum of every element in kA

Ready

Enter your 3×3 matrix and scalar k, then press Calculate.

How Matrix Scalar Multiplication Works

Scalar multiplication is one of the most basic matrix operations: you take a single number (the scalar, k) and multiply every entry of a matrix A by it. The result, written kA, is a new matrix of the same dimensions as A, where each entry follows (kA)ij = k · aij. Unlike matrix-by-matrix multiplication, scalar multiplication never changes the number of rows or columns — it only rescales magnitude, and if k is negative it also flips every entry's sign.

Formula and method

For a 3×3 matrix A with entries aij, the calculator forms kA entry by entry: multiply row 1, column 1 by k, then row 1, column 2 by k, and so on through all nine entries. From the resulting matrix kA it then derives three related quantities. The determinant scales by k raised to the power of the matrix size: det(kA) = kn·det(A), so for a 3×3 matrix det(kA) = k³·det(A) — each of the three rows being scaled by k contributes one factor of k to the determinant. The trace (the sum of the diagonal entries a11 + a22 + a33) is linear, so tr(kA) = k·tr(A). The sum of all entries is likewise k times the sum of all entries of A.

Common sources of error

  • Confusing scalar multiplication with matrix multiplication: kA multiplies every entry by the same number k; it is not the same as multiplying A by another matrix, which uses row-by-column dot products instead.
  • Forgetting the determinant's exponent: det(kA) is kn·det(A), not k·det(A) — the exponent equals the matrix's row/column count (n = 3 for a 3×3 matrix), so a scalar of 2 multiplies a 3×3 determinant by 2³ = 8, not by 2.
  • Losing the sign: a negative scalar flips the sign of every entry, which can flip the sign of the determinant and trace as well — double-check the sign of k before reading the results.

Checking your result

A quick sanity check: pick any single entry, multiply it by k by hand, and confirm it matches the corresponding entry in the displayed result matrix. For the determinant, if k = 1 the result should equal det(A) unchanged; if k = 0, the entire matrix becomes the zero matrix and the determinant, trace, and sum should all read 0.

Applications

Scalar multiplication of matrices shows up constantly in linear algebra and its applications: scaling a transformation matrix to resize graphics uniformly, adjusting the weight of a covariance or coefficient matrix in statistics, normalizing data matrices, or combining matrices in linear combinations (such as forming k1A + k2B). Because it is one of the two operations (with matrix addition) that define a vector space of matrices, it is a foundational building block for more advanced operations like matrix inversion and eigenvalue scaling.

Frequently Asked Questions

What is scalar multiplication of a matrix?
Scalar multiplication multiplies every entry of a matrix A by the same number k, written kA. Each entry follows (kA)ij = k · aij, and the matrix keeps its original number of rows and columns.
How does scalar multiplication affect the determinant?
For an n×n matrix, det(kA) = kn·det(A), because scaling the matrix by k scales each of its n rows by k, and each row scaling multiplies the determinant by one more factor of k. For a 3×3 matrix that means det(kA) = k³·det(A).
How does scalar multiplication affect the trace?
The trace is the sum of the diagonal entries, and scalar multiplication is linear, so tr(kA) = k·tr(A). Every diagonal entry is scaled by k, and their sum scales by the same factor.
Does scalar multiplication change the matrix dimensions or rank?
The dimensions never change: kA has the same number of rows and columns as A. The rank is unchanged for any nonzero k, since multiplying by a nonzero scalar does not create or remove linear dependence between rows or columns. If k = 0, kA becomes the zero matrix, which has rank 0.

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